I've owned both of those - started with a P22. The P22 needed some break-in time and a little cleanup, but was running well when I sold it to a friend of mine. It was just too small for my hands - other than that, I liked it. Bought the Mosquito after that, which fit me better. After break-in with minimags, mine runs almost 100% with Federal bulk ammo now.
They are both a lot of fun to shoot, and duplicate the controls on some centerfire handguns. The Mosquito just fits me better.
I would, for a first pistol, recommend you also look at the single action 22lr pistols, as well. Pistols like the S&W 22A or S, and the Ruger Mk I/II/II series would, I think, be a bit easier to shoot. The sight radius is longer, the single action trigger will be a bit easier to manage than the double to single transition of the P22/Mosquito. Construction is also (I think) more substantial on these. I'd feel better about purchasing a used S&W or Ruger than the other two, because I believe they will survive an uncaring previous owner better.
If you can find a local shop that has all of these in hand, it would be worth seeing how the P22 & Mosquito feel to you. Look at all their 22s, while you're there. I don't think there are any wrong choices - any 22 pistol will be affordable to shoot a lot, and get plenty of practice in.
Good luck, and have fun!