The russian government was actively looking for western professionals. An example i did research on is Friedrich August von Gebler, a german physician who got hired by the russian government in 1809. He got to know about the russians hiring people like him through newspapers (in germany the "Frankfurter Zeitung" and the "Allgemeine Zeitung"). The russian government published advertisements in these newspapers. In these ads they detailed how long you had to serve, what you got paid, possibilities for careers, and so on. When you decided to sign up, you usually had to travell to the nearest russian embassy, meet the responsible official and sign a contract (you usually enlisted for about 8 years minimum, after that you could return home or sign a new contract). You were also offered russian citizenship if you completed your contract, but that was optional. After signing the contract you were basically good to go and had to organise your travell to russia. Depending on your profession you might have to do pass some exams in one of the imperial universities (for example, Friedrich August Gebler was examined on his medical and surgical knowledge before he was allowed to practice in russia, note that this is common in almost all countries till today). Then you took up service till your contract expired.

Source: Am i allowed to quote my own research? It isn't peer reviewed or anything and was done by "hobby-historians", so i guess not?