Most shadchanim by Chasidim suggest the shidduch first to the party least likely to say yes (to minimize hurt feelings). Alternately, if the shadchan knows one side really well and isn’t afraid to offend, he’ll suggest it there first. Now here’s the monkey wrench: since generally the girls have the upper hand, they’re usually first. It’s not just a numbers game, but a quality issue. The general consensus is that there are more “good girls” than “good boys” (and that it’s easier for a girl to be considered good). Also, the concept of Nadan was left in Europe, except if the bochur is a real “illi” and his family is top notch.
Disclaimer: I haven’t been involved with shidduchim for a while so my observations may be outdated.