Do they weigh both carry on bag and personal bag for weigh limit, oe just the carry on bag. If they don't weigh the personal bag it make sense to put the heavy items in there!
I think for airlines that allow you a carryon bag plus a personal bag, they only weigh the carryon bag. So, yes, it makes sense to put the heavy items in there. In other cases, they only allow a carryon bag, no more, so you have to put the personal bag in the carryon bag. Check with your airline.
Some of the budget airlines do but in general the weight isn't a factor.
Honestly in all my travels i have only had my carry on weighed a few times and they were all when I was traveling on a small commuter plane and they were concerned about weight and balance issues.
Last New Years from Prague on Delta to New York everything was weighed and put in the sizing boxes. Then each of the carryon bags that were approved were given a brightly color adhesive strip that was checked as you boarded. When we were in line we noticed a family of 6 had a big duffel bag that was long so they conveniently placed it off to the side by some chairs and then went through and got the rest of their carryon luggage tagged. They tried to board ahead of us but the gate personal got very firm when they tried to carry the duffel bag on. They were loudly claiming that the strip must have fallen off. And the response was, Go back to the checkin desk and get another one. So do not assume anything when it comes to checking in.
Thanks for the information and experiences. I guess the best bet in all cases is to "pack light", which is what we are attempting to do. We'll be in IT GR and TR for 30 days and are carrying only one carry on bag (which is a little heavy itself as it has a handle with wheels, has a suitcase type handle and back pack straps -- weighs 7.5 pounds on its own -- and meets the h+w+l= 45" standard), plus one personal bag; she with an oversize handbag and I a small tote bag. Anyway at 73 years and 64 years we don't want a lot of weight.
Personally, I think the lighter you go the happier you will be. The key to packing light is to shift thinking from "How much can I bring?" to "Do I absolutely need this?" The former has you adding "just one more thing" because there's room, while the latter has you removing things because, the more you think about it, you don't really need them. Each person "needs" different things. I never have enough down time during a trip (even a six week trip) to read a paperback novel, while others go through several in two weeks.