I don't know about other departments coming in to ask for an engineer, but when it's an engineer looking for another engineer on a different team, it usually takes the form of asking openly in that teams hipchat/slack/irc room (as gender neutral as it gets) or specifically addressing one particular engineer (usually because that engineer is known for being great at providing advice or technical details on the issue at hand)
I've never once witnessed that second one and would be shocked if I did witness that. I've been around long enough that if that were still a problem, I would expect to have witnessed it firsthand.
I know the third happens (my sister is in sales and has been subjected to it).