From MarkoH01: https://www.gog.com/forum/general/gog_galaxy_beta_2/post5986

[overlay does not work - pops up, but input control is lost]
The Witcher Enhanced Edition

I am testing a screenshot alternative I used for Pillars of Eternity <span class="podkreslenie"><span class="bold">here</span></span>, and besides only being able to grab a screenshot with the overlay open using my alternative, I have found no problems with the overlay in The Witcher.

Here is an image of me using the overlay taken with my autohotkey + imagemagick script. I had to reduce the size by converting to jpg in order to preserve the quality and post in the GOG forums.

Since my alternative screenshot functionality is very lacking, I hope they fix it before they chose to move Galaxy out of beta. At the least, I no longer need to alt-tab mspaint my screenshots for games that allowed it.

Added:
[overlay doesn't work - from omega64]
Metro 2033 Redux

Moved from doesn't work section to works section:
[overlay works - kittehoverlord]
The Witcher Enhanced Edition


Note: For the time being, games that the overlay appears to work ok for at least one person are currently being listed in the [works] section. If someone tried a game and it didn't work previously, they might want to retry it again to see if it works for them now. Also feel free to update with "still doesn't work for me" or similar. Down the line I'll probably post percentages for works/doesn't work.

[overlay does not work]
Wasteland 2 Director's Cut Digital Deluxe Edition

I was bored so I tested some more games ;P

[overlay does not work]
Painkiller Black Edition
F.E.A.R. 2: Project Origin + Reborn
F.E.A.R. Platinum
Deus Ex™ GOTY Edition
Blade of Darkness
Chronicles of Riddick: Assault on Dark Athena, The
Freedom Force vs. the 3rd Reich
Giants: Citizen Kabuto
Luftrausers
Painkiller Black Edition

A couple of these aren't listed on GOG anymore, though.

According to GOG, FarCry 2: Fortune's Edition now works with the overlay

Dying Light - The Following - Enhanced Edition

I had a very strange effect with this game, and had to disable the overlay:
When running (holding Shift) the the character started to look in random directions (like someone was moving the mouse to look around). Which, in this game, is a pretty deadly effect (running across a beam being pursued and suddenly looking right and up = *splash*)
I thought at first the game was bugged, disconnected the controller and tried googling this problem (with no results that told of similar behaviour).
Finally I had the idea to disable the overlay - et voila - game behaves normally again...

Interesting. As I was playing Wasteland 2 with my controller, I would suddenly get random inputs. Like I would press down on the left stick and it would act like I hit the left bumper, ect. I believe this happened after I tried to use the overlay, but the overlay doesn't work for WL2...

[overlay doesn't work]
Thief Gold
Outlaws
Pier Solar

[overlay does work]
Big Pharma
Virginia

Updated, thanks.
Thanks for reporting guys, this means that the overlay is technically working but the code that controls capturing the keyboard input from the game is failing somehow, or in short - it is buggy. Please report the bug in GOG's bug tracker at http://mantis.gog.com so they can look into it and hopefully fix the problem for these games in the next release or so. Be sure to mention what games are affected by the input problems and as much detail as possible as it could help them fix it. Does Galaxy have a checkbox to disable the overlay for these games since the 1.2.5 release? I noticed some games have it and others don't oddly enough. :)

Added Firewatch to the "works" list based on https://www.gog.com/forum/general/gog_galaxy_beta_2/post6111

Works:
Bit Trip Runner

Does not work:
Tower of Guns

Thanks, updated.

Saints Row IV: Game of the Century Edition and Saints Row: Gat out of Hell both work with the overlay as reported by ElTerprise

Prince of Persia: Warrior Within - works, reported by Random_Coffee