I think it's OK as a buyer to leave a "kindly reminder" to ask for feedback from a seller, provided you have already left feedback for them. I get feedback request from buyers all the time. I don't use auto-feedback, so often a buyer will send a message kindly asking for positive feedback in return.
As a buyer, you have a bit more leeway - I wouldn't exactly recommend begging, but most sellers are unlikely to get upset if a buyer requests feedback once or twice. If you don't get any after that, I would probably just leave it alone.
As a seller, it's the right thing to do to leave feedback for a buyer who has left if for you. It is voluntary, however, so you don't really have grounds to complain if the seller doesn't.