Pretty broad question. Yes is the simple answer. What specific things depends on the data and organizational relationship.
As a customer or authority, I'd be insisting on certain minimum integrity checks before it was submitted to me. That might require working out schema before hand and/or writing custom tools. I might also QC/QA spot check their submissions with some of the same tools or checks, or even some of my own that I don't necessarily give them.
If we're 'equal' or cooperative departments, then there's less I can do from a demand standpoint and more about working together and understanding each other's process to improve both sides with minimal extra effort. I wouldn't expect them to do some extra involved process that only benefits me and my use of their data.
And of course, as Michael mentions, good/accurate metadata/documentation is always good.