SF: Approaching is fine. Going on a date just because someone approached you is not fine.”
I think approaching someone (in a respectable place and in a respectful way) is perfectly fine. I FURTHER think that if the person would like to go out with you (after her ascertaining from a decent amount of conversation that you are probably not an ax murderer or something), that it would be ok to go out for coffee or the like and thereafter make your inquiries. Both parties should keep it VERY public, and not give out more personal info than that with which you are both comfortable.
I have made no secret of the fact that I do not believe that everything has to always go through a shadchan, necessarily. People who are mature enough to be getting married, SHOULD be mature enough to assess whether or not they want to go out with someone whom they happen to meet. Sometimes THAT is the bashert thing that is meant to be (Yaakov and Racheyl meeting at the well) and happen in that way. We don’t always get a second chance.