spoilers:end of spoilers
Thinking about it, it's plausible for Peter to react as he does. He doesn't have combat experience; and the first reaction of any normal person who doesn't have combat experience to a conversation is going to be to continue the conversation. It would take time for a normal person to learn not to drop their guard when someone tries to talk to them.
But, yes, apparently no spider-sense. But spider-sense is always a bit inconsistent. It makes life difficult for writers, since our hero getting surprised or ambushed is a standard way of generating drama that they can't use if Peter's spider-sense is functioning.