The choice of time and place is irrelevant when the instigating party said, and I quote, "anytime, anywhere", and then when their challenge was accepted, stood up, ready to engage in said challenge.

In response to people saying that the Senator acted inappropriately. But perhaps I misinterpreted. Are you saying they both acted inappropriately? In which case I agree, but I'd still say tweeting a challenge is less inappropriate than actually accepting that challenge at that time and place.