When using upstream's bundled wine, a user is able to share their screen, however, currently this does not work with native wine. Just a side note, in both cases, the user is able to view a shared screen.
That's not the only glitch with system-wine, which is why the previous maintainers had it even masked afair. 'system-wine' should not be default.
The problem (TV client hanging at "initializing display parameters") still persists with net-misc/teamviewer-11.0.53191 and app-emulation/wine-1.8. I'll try to resume communication with TeamViewer GmbH initiated by NP-Hardass regarding this issue (I'm also a paying customer, so let's hope for the best ;)).
This version of net-misc/teamviewer was removed in git commit 10c2ceaeed402f8f73447dbac9a1bfceb00aef02