It IS possible to buy a piano privately; however, if I were you, I would at least find a friend who knows how to play to come and take a "test run" for you. If you would like to find out whether it needs to be tuned, a good way to test it for yourself would be to buy a pitch pipe of your own-- one that's circular, with all the notes printed on the top of it-- you know, A, B, C, D, E, F, and G, with all the sharps between each note. When you test the piano out for pitch, the "C" note is the white one directly to the left of the two black keys together. The notes (starting with that white one) go C, D, E, F, G, A, B, then another C, another D, and so on. While it IS possible to buy from a dealer, it is EQUALLY possible to buy from a seller privately; in fact, you're more likely to find a great deal buying that way. If you decide to find one yourself, there's Craigslist, Amazon, and a ton of other ways to buy privately; however, if you decide to do that,, then at least find a friend who plays him/herself so you can get someone who knows how it feels to play it-- that way you can tell him/her what you would like to get, feeling--wise, from the piano-- like do you want a clear tone from a light touch, or one you must hit each key harder, or something in between? (For instance, I like a relatively light-touch piano, while my sister, who also plays, wants one where you must hit each key somewhat harder.)
It IS possible to buy a piano for under $800, though you might have to look in the classified ads of your local paper, or go to the local college or community college for their yearly piano sale.
The fact that you would like your child to learn alongside you is wonderful! That way, you can grow closer to each other both musically and in your relationship! While my parents did not play, my grandfather was a piano teacher, and I still cherish memories of time I spent playing alongside him. Hopefully you will grow closer through your music! Good luck!