Fairly sure you'd need to check both universities policies. Sometimes they have clauses that say you can not be sutyding in two places at once without express permission.
I think also, to study at another university, your degrees must relate. For example, I'm doing accounting, & If I wanted to do a sub major in Tax Law, my uni doesn't offer this. I know UTS offers this, so I could be able to apply for permission to complete the tax law sub at UTS and receive credit for this when I graduate at UWS.
Because I'm doing an accounting degree though, I wouldn't be able to study, say, Philosophy or Medicine at USYD because its not related.
I think the best option though would be to speak to both, because it would suck if you weren't allowed to and either one of the universities or both kicked you out because of it!!
And as the other guys have said, the clashes would be hard to avoid. Most unis have policies that say if you have an exam somewhere else you can defer it and you have to do theirs. Would suck to have 2 exams with each uni demanding you do theirs or fail!