Is it possible to send same message to a group of users in at one time?
Are there any tools which could do that? I understand such function can be misused, but also could be very useful in if used in a correct way.
Not really. In fact, due to abuse by spammers, there is limitation on how many messages you can send (this refills over time and is higher for more active accounts).
How many messages you need to send? Maybe inviting this people to some chat group on one platform or another would work.
As has been noted there is no API or similar mechanism for this, the current rate limit for sending messages manually is ~50 messages/hour.
It should be pointed out that sending unsolicited messages is in general a touchy subject. Currently there is no way for OSM account holders to indicate if they are OK with that or not. As a result you should use it very very sparingly.
I would note though that Communities - communities by openbrian · Pull Request #3717 · openstreetmap/openstreetmap-website · GitHub would provide at least a partial solution to the problem, if it ever progresses to a state in which it can be merged.