General practice is that the suggestion goes to the man’s side first. The easiest explanation for this is that if the man isn’t interested, the woman never has to know anyone was suggested and declined. That way, her feelings are spared. This isn’t to say that women are too weak to handle rejection. It is just a fact that hashem created women as more emotionally charged than men. Therefore, the nature of man, in general, is to deal with rejection and disappointment easier than women.
It is not halacha or in any way imperative that the man hear of the shidduch first. In some cases, when there is an extenuating circumstance, the shadchan may chose to speak with the woman first. Still, my recommendation is that the shadchan should ask the woman questions to get the pertinent information without mentioning the specific shidduch. In other words, if a shadchan isn’t sure that a woman is okay dating a convert, the shadchan doesn’t have to actually make the suggestion, she can just ask the girl if she would be open to a convert. In either case, the same protocol is followed.
Among young FFBs, the suggestion will often go through the parents. Among BTs or older/secondtimers, the suggestions are usually made directly or through a dating coach, friend or Rabbi/Rebetzin. It is always safe to call the single directly to ask them if they want to hear the suggestion or have it go through someone else. If you are the one making the suggestion, it is nicer to make the suggestion to the guy first. This is not a matter of halahca, it is a matter of protecting the woman’s feelings. If the guy says no, there is no reason for the girl to know anything about it.
I always suggest to women I work with that if someone calls them with a suggestion, they should ask if the guy has already said yes. If he hasn’t, I recommend that the woman kindly say that she prefers not to hear any details until the gentleman has already agreed to the shidduch. This is so acceptable, that some dating sites even have an option in their preferences that the woman gets to chose if she gets a suggestion first, at the same time or only after the guy has said yes.
When the shadchan calls the man, it is up to him to tell her if he is “busy” with someone else. If he is not busy, the shadchan tells him about the girl. At this point, the man is welcome to ask whatever questions he has about the girl and the shadchan gives him the girl’s name and the names and contact information of his references. This is the proper time for him to do whatever data gathering he chooses. Once he accepts the shidduch, the shadchan then presents it to the girl. At this point, the girl can do whatever research she wants to do.