You could be a:
I can be the life of a party,
or just for you all alone.
You can play a record to listen to music by yourself in your room, or use it as the soundtrack to your social occasions.
I may be 2 faced,
but when you use me I'll show you only one.
A record has two grooved sides (faces), but only one is in contact with the stylus of a record player while it plays.
Anyone can own me,
but most must pay a price.
Anyone can own a vinyl record. In most cases this will be by buying it from a shop or seller (paying a price).
Damage me and I'll sing & skip,
but if you break me that's when I'll stop playing!
If you scratch a record (damage it) the stylus will skip or get stuck, interrupting its 'singing'. Break it in two and you won't be able to play it at all!