The second sentence is better but not perfect. I would suggest that you use assumed instead of supposed.
If this is part of a letter or communication that is even slightly formal (and in the context of dunning letters it is probably safe to assume this) it is better to use we have instead of we've. Also, it is on you is an informal, if not quite slang, way of saying it is your responsibility.
I would suggest rewriting the sentence as follows:
So far we have assumed that it is your responsibility to check whether to send dunning letters by e-mail or by mail.