@LookAlterno hey! Do you own the room?

As far as I remember, you must be the owner of the room to invite someone. If you own the room and still cannot figure it out, let me know and I can help invite you

If you do NOT own the room, I would advise that you first talk to the room owner. If there's some special circumstance/the owner isn't active anymore a moderator has the ability to help, but most wouldn't due to it being a private room

@LookAlterno See Why are there no private chat options? - it seems that Stack Exchange does not support private chat rooms, other than for moderation purposes. We moderators have a private room where we can discuss confidential issues, but that concept is not offered to non-moderators. Sorry.

The other day, a new user invite me to talk to clarify some matter. I still can't figure out how to do that. Shame on me.

you can invite people in a chat room to a new chatroom, click on the name and you should see the option

It's common on forums (including the Arduino forum) for people to want to get private coaching. They latch onto someone who looks knowledgeable and try to get you to solve their problem personally. If you fall for this you will end up spending hours to-ing and fro-ing with every tiny detail. This site isn't designed for that, it is supposed to be a knowledge-base. So, private conversations about your problem are not encouraged.

Just say "ask your question publicly, then everyone (including people in the future) can benefit from my answer".

The other problem with private (technical) conversations is, if you give some bad advice (quite possibly inadvertently) then no-one else can correct that, since this is a private chat.

There are places for "chat" style Q&A sessions, the Arduino forum being one of them.

a chatroom with more than a few regulars will fix a few of these issues but asking on the main site is still better in general

Another possible problem is people trying to get you to do their homework for them. If you answer publicly (as in, a normal question) then the teacher may well find that their homework has been posted on Stack Exchange, and that the answer given is exactly the answer their student gave to them. By wanting to go to private chat the student is trying to bypass this problem (and get a good mark based on your knowledge, not theirs).

I've never been a chat guy, anyway. I prefer email, which let me express myself better.