Ferd's method works, assuming you can actually manage to help with the dishes (the trick to that is to just start doing it, rather than ask... if you ask, the host is obligated to say "no, of course not," since it is understood that you don't actually want to help with the dishes), but the one I had in mind is you take a serving implement, pick up the last piece of chicken, catch the eye of someone else at the table, and offer it to them. They, of course, are obligated to say "No, you take it" (as are you, if someone offers it to you). If you are a guest, or the youngest person at the table, it's OK to accept at that point. Otherwise, you can look around to the table -- with the chicken still on the serving implement in your hand -- and ask if anyone else wants it. They all say "no," of course. Then you can serve yourself.

Which was all way too complicated to explain to someone who was having trouble with the idea that "Oh, can I have the last piece?" was rude by local standards.

Volunteer to help with the dishes, then ask whether you can take away the plate the chicken is sitting on. If nobody else claims it, it's yours.

Clear another plate before you touch the one with the chicken on it. Clear something else after. Clear your plate when you're done eating.

Don't do more work than your hosts. You're being helpful, not trying to work off the price of dinner.

FWIW, among my friends--whom I might describe as "polite askers" or "assertive guessers"--it's common to ask "does anybody want to split this with me?" That way, you're both asking for what you want (more of the thing) and making an offer in a guess-culture-compatible way. It's easy for other people to accept, because now by taking it they're not preventing you from having it. If no one does, you can be reasonably confident no one else actually wanted it.

A variant on the same thing is: "Would anyone else like this?" which is a shorter version of the offering ritual that TheOtherDave described. Because it's skipping most of the ceremony, it's much askier, but it's still not polite to say "yes" and take the thing, because you'd be taking it out of the hands of someone who clearly wanted it. (An exception might be made if you hadn't actually had any of the thing yet, and said so.) But you can say "I'll split it with you," achieving the same result as the above.

Of course, this only works for plausibly divisible things. I've had a friend laugh at me--good-naturedly--for offering to split something bite-sized. Surprise, surprise: he's much askier, I'm much guessier.