what kind of local specialty can we take back in France?

If you are talking 'local' you really need to specify which local part of France.

I'm traveling in Paris.

Wine cheese (France has over 500 varieties!)
Uh but you should be mindful of the US government restrictions on what you can and can not bring in
And check it in your luggage. My sister bought a little jar of special made honey in Provence and they confiscated it in security because we had no time to go back to the check in counter.

That's right! I should be mindful of the restrictions. I forget it and thank you for your remiander.

We brought back some Le anis de Flavigny candies as gifts. They come in small tins or boxes and don't take up much room in luggage. We bought them in Burgundy (where they're made), but they are sold many places in Paris. We also brought back small jars of Edmond Fallot mustard. It's wonderful.

Where are you going back to?
US Customs, Australian Customs, etc all have different restrictions.

The US coustom will be went through.

Truffles, is it easy to bagged my luggage?

I recommend sea salt as that is least likely to be taken at customs. Hard cheese might be ok but do read the facts from the government about what is legal. Mustard is a good idea too. Spices such as herbes de Provence is great. Food markets sometimes have bagged spice mixes for little money. They are light to pack and dry so spices and specialty salts are my go to gifts for family friends and me!

I read an article online yesterday about shopping for gifts from Paris at Monoprix; singly packaged soaps, the brand may be petit Marsailles (spelling likely wrong) made in France, salts and mustard was on their list. There are numerous scents of the bar soap. Pretty tea towels and clothes too at Monoprix. I have a hand towel bought from a tourist shop that is round with a fabric hook for hanging; my mom uses hers daily. A bit bulky but a nice reminder with Paris stitched on it.

I usually buy bookmarks at museums for friends and family as we are all readers. Tiny and light to pack. I have an Eiffel Tower bookmark any my desk at work which I gaze at longingly often.

Scarves and more scarves! I am a scarf-a-holic! I bought a great scarf at Monoprix that I wear a lot; they have a good selection. The open air food markets often have scarves to sell. Last year my favourite scarf was purchased at Galleries Lafayette for $20 Euros.

Enjoy shopping in France.

https://www.cbp.gov/travel/clearing-cbp/bringing-agricultural-products-united-states

Alcohol and tobacco are a matter of volume and tariffs.

Like June, the gifts I most often bring home from Paris are bars of wonderfully smelling soaps and sea salt. They always seem to be a big hit with friends and family and are easy to pack.

Chocolates, Angelina hot chocolate, go to le grande epecerie paris. It is part of lt bon marche department store. They have lots of really interesting foods and other gift type items.

Any packaged soft cheese like Brie is OK, no problem. For wine they do sell sealed and padded plastic carriers for checked baggage for $2-3, but cheaper in US before leaving. Everything soft or liquidy should be checked. Chocolate can melt en route and the expensive chocolate we brought home was kind of nasty so taste beforehand.

If you buy cheese or butter, have the merchant pack it in vacuum packs and put in your checked luggage where it will stay cold for the duration of your trip. This goes for chocolates, too.

We bring back canned duck confit because we can't get it in the US (you can get fresh made duck confit in the bit city where we live but not the canned kind which is frankly the good stuff) This once triggered a full customs search; we had indicated 'commercially canned duck' on the customs form, but they didn't take it. The next time the agent didn't care.

It becomes increasingly difficult to find things anywhere you can't get on line anywhere. On our last trip I bought a couple hundred Euro in children's and baby clothes at Dpam, Sergent Major and similar moderate priced shops. They are cute, cost effective and different from what I can buy locally and the kids love them.

Our favorite was the Sea Salt. (Fleur De Sel) and we bought it at Monoprix or Franprix.

We also took back some Macarons in different flavours. I wish we would have made it to a famous place to buy them but we just got them at a local place.

Tea Towels are nice and some soaps.

My hubby got flavoured roasted pistachios at a store across the Seine from the Eiffel Tower that he brought back and LOVED.

Happy hunting.

Janettravels44 has the right idea. The internet shopping experience has greatly simplified my purchases abroad.

I can purchase Dijon Mustard https://www.amazon.com/Amora-Dijon-Mustard-Pack-Large/dp/B0085VZFA6/ref=sr_1_6_s_it?s=grocery&ie=UTF8&qid=1497913459&sr=1-6&keywords=dijon%2Bmustard%2Bfrance&th=1

Sea salt https://www.amazon.com/Saunier-Camargue-Fleur-Ounce-Canisters/dp/B0002FGY9O/ref=sr_1_3_s_it?s=grocery&ie=UTF8&qid=1497913522&sr=1-3&keywords=french%2Bfleur%2Bde%2Bsel&th=1

French truffles https://www.amazon.com/Guillot-Summer-French-Truffles-Choice/dp/B000LR0P9I/ref=sr_1_1_s_it?s=grocery&ie=UTF8&qid=1497913584&sr=1-1&keywords=French%2Btruffle&th=1

Duck confit https://www.amazon.com/Rougie-Confit-Duck-Legs-52-91oz/dp/B0008JGWB4/ref=sr_1_1_s_it?s=grocery&ie=UTF8&qid=1497913668&sr=1-1&keywords=duck%2Bconfit&th=1

Foie Gras (goose) https://www.amazon.com/Katealde-Artesanos-Bloc-Foie-Gras/dp/B018SPU3V4/ref=sr_1_1_s_it?s=grocery&ie=UTF8&qid=1497913747&sr=1-1&keywords=foie+gras+goose

And even Escargots https://www.amazon.com/Escal-French-Burgundy-Escargots-Snails/dp/B00TLS5K4I/ref=sr_1_2_s_it?s=grocery&ie=UTF8&qid=1497913866&sr=1-2&keywords=french%2Bsnails&th=1

Importing wine is heavy, cheese is subject to Department of Agriculture restrictions. Little trinkets make better gifts, IMHO, and are generally lighter and smaller, all the better to fit in your carry on.

I tend to buy tea towels or a coffee mug for myself on trips. The thing with the food items is that there is almost nothing that you can't get where I live. It is customary to bring back some candy to share with your office. I live in NYC and people travel a lot.

We can probably buy 50 kinds of sea salt including the pretty crystal flakes where we live. We can buy hundreds of scented soaps although we sometimes do bring back the cheap Monoprix available soap as we like the smell. Foie gras and confit CAN be bought on line but it is really dramatically more expensive (which is not the case with salts and soaps)

Macarons travel very badly and are increasingly available in the US; I can buy bakery made macarons within a mile of my home. Trader Joe's frozen macarons are about like Piccard's frozen macaron; not as good as those made fresh at an artisanal bakery but better than the fragments that are likely to survive travel or macarons a day or two stale -- they are pretty good. Great chocolates are also easily available in the US including Maison du Chocolate candies which are excellent but there are brands in Paris not as easily acquired at home. I can't take them in the carry on as they oddly disappear in flight. A very expensive, very unique bottle (or case ) of wine might make sense for an oenophile but for anything less it is a lot of hassle for something you can get 100 varieties of at home.

One thing I always bring back are the little plastic apples filled with gel that are designed to cut smells in the refrigerator; I can't find those in the US or on line in the US and they are so handy in Paris when we have cheeses in the refrigerator and equally handy at home. They work better than a box of baking soda. There are also a couple of brands of hot chocolate that are superior to what I can buy here, although we have a Girardelli store 6 blocks from us and their powdered hot chocolates are almost as good.

Importing wine is heavy, cheese is subject to Department of Agriculture restrictions.

We do bring back anywhere from 6-10 bottles of wine, but yes, it is heavy, and good packing is important. We also bring back lots of cheese. We've never had a problem (we always declare it), and if you read the government info, there are very very few restrictions on bringing in cheese for personal consumption (commercial importation is different); raw milk cheese is generally fine, even if not aged. It's really only liquidy cheeses (ricotta?) that are a problem.
We often bring back chocolate; just bars, not special chocolates from a chocolate shop. We recently brought back duck rillettes, and the Customs inspector didn't flinch when I included it in my listing (maybe he didn't know what it was). Honey can be worth bringing back, but potentially messy, and would most likely have to be checked (we did so).