You are from the land of Chopin, and you're playing Skynyrd?

Yeah - we're in music classes every day, but only lessons for an hour each week (typically). At my school, we also meet once a week with all the piano majors and faculty for a recital class, where we play for each other (it's not required to play unless you're a performance major, and then only once a semester). Once a month, instead of the piano class, the entire music school gets together in that time period for the best of the departments. My studio is a group of very independent folks so we're lazy about studio classes, but some piano studios have a required studio recital class for an hour each week (there are four piano faculty - two of them require it and mine has an optional one - the teacher is always there, so it's like a free lesson if no one else shows up, on the nice Steinway).

Back to being independent...there's generally not much harm in being independent because your teacher will generally be able to pick out bad habits even if you're developing those bad habits by playing music that's different from what she assigns you. If she assigns you a 3-part invention, then practice it and when you get bored, sight-read the other ones. Polish the ones you like a little bit. And play some Chopin. It's good for the soul. If you're feeling really wild, get a Real Book or two and try some jazz charts.

My tried and tested advice - play Bach. Only if you like it, of course, no point in going against the grain. Also, play Bartok - Mikrokosmos and For Children. Don't be fooled by the titles - this is real music ranging from very easy to quite a handful. Try to play all these best as possible, and you'll be a good pianist. Don't be obsessed by difficult music just because it's there. Simpler things can be just as rewarding musically, and be just as much of a challenge as there's nowhere to hide, and there's no excuse for fumbling.

I think it's ok to practise on your own, but do let someone else evaluate your playing from time to time.