Are there any screen sharing apps that are more geared towards maintaining privacy?
I'd suggest using an audio call (phone, Skype, Google Hangout) and a code pad that is synced in realtime (free ones exist like collabedit.com). In other words, don't share your screen, share your voice.
Use a fake name/alias. Sign up for something like Cloud9 (https://c9.io) or use another synced editor like collabedit or even Google Docs (make sure to use the fake gmail account and not your own).
Sign up for Skype with the fake gmail account and fill out the details like you did on the fake gmail account.
Then sync up with someone online using audio only and using the synced editor of your choice.
You will leak your IP information most likely so they could figure out in what vicinity you live, but you could setup a proxy if you wanted even more security to hide your location.
As long as you're not hiding from any three letter agencies this is probably more than enough.