Tipping a private driver?

You mean European customs are different. The driver does not expect this but you can slip her or him a 2 euro coin for each bag handled. Just don't make it awkward by fishing around for it so she he can promptly go about their business

We tipped our driver for this same trip 20 euros. He was so wonderful, informative and personable, plus after seeing the conditions in Naples just getting out of the train station it was worth ever cent of it. If you need a recommendation for a car service let me know.

This might help you
https://community.ricksteves.com/travel-forum/italy/tip-or-not-tip-a-private-driver

An italian wouldn't tip at all and drivers do not expect tips from Italians. A private driver's job is driving and helping with bags: the idea of paying more for something that's always been included is quite ridiculous for italians.
Americans have been tipping for ages and now drivers expect you to keep on tipping. If being on the wrong side of double standard pricing doesn't bother you, keep on tipping. I'm sure the driver won't complain and the number of italians using private drivers is small anyway. Once you know you don't have to, you can decide by yourself.

The area around Naples Centrale station is the city's worst area, it's not an example of the all city.

You don't need to tip for this type of service or for anything when in Italy. It's not a tipping culture. If you're out to dinner and want to "round up" the bill that's one thing but "tipping" as is done in North America isn't necessary. You won't see Italians giving the driver anything beyond the agreed to fee.

Donna

we used two private drivers, last week,on the amalfi coast. we tipped both, a couple of euro each person because one driver took us very carefully from sorrento to amalfi ( not driving like a formula one driver) and showed us places of interest. the other gave us a fun time explaining all the points of interest. neither of them expected a tip. so we didn't mind giving them a small token.
whereas some restaurants in the area ( by comparison) tell you no tip is included!!! when they hand you the bill.

hope this helps

See this recent discussion on the Forum about tipping in Italy. This topic comes up frequently in the RS Italy Forum. There are two camps: (1) those who never tip in Italy; and (2) those who tip in Italy. I suggest that you read the arguments for and against and then make your own decision. There are many other discussions pertaining to this topic if you do a search, but I was a participant in the cited one, so that is why I suggested it.