One-time pad means that you have a limit to how much data you can exchange before the pad is spent - if you send enough messages, then you can't send any more securely until you exchange another one-time pad.

You're not likely to get a collision ever because of the typical sizes of one-time pads - birthday paradox matters if the key length is measured in bytes, but not if it's measured in gigabytes, there isn't enough space/time in our universe to expect a collision.