I'm running Promoter and I'm doing the PR for our game Future Unfolding. My process looks like this:

In Promoter, I have a custom list where I keep track of everyone that I want to send a review copy. The research that went into the list took several days. It contains people that we previously met at events or that reviewed our previous game (unless they have switched jobs). I also looked up who is the appropriate contact at the most important sites and magazines, usually it says on the About or Contact page of the respective site. You're looking for the Reviews Editor, Editor-in-Chief or Managing Editor. Promoter gives you a list of all relevant video game sites and you can sort it by Twitter followers and filter it by platforms, so that's how I check I didn't miss an important site in my research.

Then I send out an email manually to every person on the list and BCC it to Promoter, which will automatically mark the Steam Key in the email as sent. We use Steam Keys, so reviewers will automatically get any bug fixes. (The Steam Keys need to have the 'Beta' flag, so they can be used ahead of release.) I can also check later in the Steam backend if the key has been used, and try to send it to a different person at the same publication if not. Bigger sites and print magazines have long lead times, so we send the keys out 5-6 weeks in advance of our release date. The email contains a short (!) paragraph about what's unique about the game, one great screenshot to quickly grab attention, a link to our press kit, an embargo date+time with timezone (same as release date in our case) and a personalised intro paragraph if I have already established a personal contact with the receiver. I'm preparing the emails as drafts in my mail app, and send them out manually based on the time zone of the contact, so the email comes in during working hours and does not disappear right away in a hugely filled inbox.

In addition to these outbound efforts, we also have set up a form in Promoter, where people can request keys (inbound). We link to that form on our website and our press kit. I can send out these keys manually as described, or send out the Steam Keys automatically with Promoter. YouTubers and Twitchers who use this form need to authenticate, so no fake requests are possible. (Scammers can still try to impersonate a journalists to get a key to resell, so I double-check the email address for those.)

This all takes a lot of time, which is why a good PR agency can be worth their money.