If you have not done so already, please email supportdos@larian.com, with a description of the problem and the report.zip file generated by the
D:OS support tool (for the PC version) for both systems. The report will contain system and game information, as well as your saved games. To reduce the file size, the host can delete all but one save from the zip, and the person joining can delete all saves (if they have any from a different playthrough, etc).
Have you tried switching who was hosting a game?
Are you shutting down all non-essential programs (especially anti-virus) before starting the game?
Have you tried the Steam invite function? One person during the beta had problems joining a game from the loby list, but if they were not in the game when receiving an invitation the game would load and join automatically.
There is also Direct Connection, using either an IP address or the Server ID (that the host can get after loading the game, in the Show Connection menu [diamond icon top of mini-map]).
Can you try a LAN connection? (possibly combined with VPN software like Hamachi or Tunngle)
If anyone is using a laptop, set the power options to high performance and make sure the wifi is not set to turn off after some period of inactivity (someone reported that that caused frequent disconnects for them, though they were able to connect to start with).
Depending on your modem and connection method, can you bypass your router and connect the modem directly to your computer? Port forwarding may help, since this seems like a networking issue. Copied from another topic:
The game's client will try to assign a port in the range 23243-23252, the server in the range 23253-23262. While forwarding these ports on your gateway might help, there is no guarantee the gateway will assign an external port in this range when connecting to our NATPunch server (this is the port that will be assumed to be the external port for all connections).