Do I need permission from the other home owner to pave the driveway that we share?

You very well may need their permission, but first, you need to determine ownership and easements. For example, if you each own part of the driveway and have an easement to use the other person's portion, you would need their permission--you can't pave someone else's property (e.g. the 1/2 they own) without permission. If you own the whole thing but they have an easement to use it, you can pretty much do what you want as long as you do it in such a way as to not--or only minimally--disrupt their rights; in this case, you're the property owner. If they own the driveway and you have an easement, you can offer to pay to pave, but it's their property entirely; you have no rights other than use. Etc.--you need to determine the exact situation before determining what you can do, and every shared driveway differs. For example, my current home as a "shared" driveway that is actually two narrow, side-by-sides with a reciprocal easements to use. A home we locked at purchasing had a "shared driveway" that it owned entirely, but the neighbor had an easement to use it.

Note: shared driveways are a pain, as you are finding out. If there would be someway to reconfigure and create your own driveway, even though it would be more expensive, you may wish to do so. Again, in my case, we've been unable to do other than patch our own section for 5 years since we can't get our neighbors to agree to larger maintenance and repair.