There are a ton of avenues you could explore.
One would be modifying the events query using the provided actions and filters; another could be to build a remote API around The Events Calendar on just one of the networked sites, then use that same API to pull in the events on the other sites.
Really you’d need to do whatever you/your developer(s) feel comfortable with – I did chat to one of our devs about this and our feeling is that this is unlikely to be taken on board as a feature request, but it would definitely be great to hear back from you as to how you go about solving the problem 🙂