I have played many PbPs, including those with Paranoia.
One of the downsides with text form, especially with Paranoia, is it's harder to get away with spelling based gotchas.
However, it's fairly easy to run in a PbP, especially as a "play when ready" scenario.
Regarding your specific question, re "info leakage" you'll have to make the Sheets private, but encourage your players to perhaps "overshare" to compensate for this.
Don't forget, Friend Computer knows all and is an excellent storytelling device.
New Message from Friend Computer: Someone was seen wearing Commie iconography near where Player2 was last seen
Another easy way to allow this leakage is to create different "rooms" on your server for each area so players discussing something in the mess hall can be "reasonably" overheard by another player that happens to be there.
I would also recommend using Zap rather than Classic, as PbPs tend to be rather slow, even with interested parties present. Zap can mitigate this somewhat.