I'm almost positive that nothing even as 'temporary' as a shed can be put so close to the property line, let alone ON a shared driveway. In my municipality, a shed is not a permanent structure. So, it is governed differently than, say, an addition to a home (which IS permanent.) Since a shed in my township isn't permanent, it can be put anywhere on my lot BUT no closer than 35' from the property line. (If an EXISTING shed is being replaced with a shed of the SAME SIZE, or smaller, and the 'old' shed happens to be 12" from the property line, I can put the NEW shed right where the OLD shed is.) It can get comnplicated. (If I was putting up a BIGGER shed then the one I have, or putting up ANY shed where I never HAD a shed, I need to follow the 35' Rule.)
Your Easement should state the use of the driveway if I'm not mistaken. If you used a Real Estate Attorney at your closing, run this by him/her. (You did have one looking after your interests, didn't you?) Once you get your 'answers', talk to your attorney as to how you (or he) should approach your neighbor. It's a touch situation, I know. You want to be neighborly but you want 'the law' to be followed. (Otherwise, what good do they do?!)
There is a legal expert on the Forum. If our Expert gets a minute, you may hear directly. My best to ya and hope this helps.