I had an "incident" a while back where I was told to leave DS standing alone, and I flat refused. (This was at DEN.)
Normally, our rule is that one of the adults goes through first, and after that person clears completely through, then the other parent will send the bags through and then guide the child. That way, if one parent is pulled aside for a detailed search, the other parent will be able to remain with the child.
In this case, there was no line at security, and the officers INSISTED that I could not hang back until DH was completely finished; they seemed to think that was suspicious behaviour. Sure enough, he got pulled out for a detailed search behind the barrier. As I went through with DS, DS cleared with no problem, but they wanted to search me. I told them politely that I would be happy to cooperate just as soon as they cleared DH, or that they could do the search with DS present, but that I would not leave my child unattended. They threatened me with arrest if I did not come immediately, without DS. Luckily, DH was cleared through just as they were calling someone to come arrest me for resisting the search. It turned out that the issue was the underwire in my bra.
BTW, if the stroller will fit through the scanner when folded, you will be asked to fold it and send it through. If it won't fit, they will wand it, but anything hanging on it or in the basket will have to be taken out and scanned separately. Bottom line is, get rid of all the bits and pieces of metal before you reach security, either on your selves or on your stroller. I slip my shoes off and send them through in the carry-on; less hassle.