Currently there's no way to have a private game with friends, no custom matches, no bots. These are not the only issues with the multiplayer in Doom, but these are on id's radar to fix.
In an interview with Eurogamer, id's Marty Stratton outlines the changes coming. Private matches and custom game settings are coming later in the summer in a free update, but bots will take a bit longer to roll out.
Read the full details on Eurogamer: http://www.eurogamer.net/articles/2016-06-09-id-takes-back-control-of-doom-broken-multiplayer-on-pc
EDIT: id has already released a server-side multiplayer patch that addresses some multiplayer issues. (Thanks to Greg for the tip)
And in other update news, a campaign related patch coming later in the month has some neat changes, and Bethesda has announced four of them:
- Added Photo Mode for taking in-game screenshots
- Added Classic DOOM weapon placement/view model option (1st person view option)
- Added new Snapmap features (including skybox windows) and AI pathing and logic fixes.
- Fixed various bugs including players accidentally triggering Dev Mode and Trophy/Achievement bugs (IDKFA, Every Nook and Cranny).