In my non-expert opinion, the answer would seem to be yes. I can't imagine any security company wouldn't allow direct access to log-in and check your cameras, history, etc, whenever you wanted. I would think that would be downloadable as well to your personal computer, for example, while the security company would maintain the original digital file.
The only limitation I could see would be if you're using a phone to check. I think he would have been able to screenshot on his phone while watching the video, but that would probably be as much as he could do on a phone. So maybe he screenshotted the video while viewing on his phone and gave those to police.
I could be wrong, but I can't for the life of me think that a security company would prevent a client from downloading footage. So I'd have to think that is possible.