Well, first you'd have to decide whether you're ok with taking notes on pen and paper. Many law students prefer taking them on their laptop. This is a matter of preference, and you're not at any disadvantage for taking notes on pen if that is what you are comfortable with.
The vast majority (at least here) do take their exams on laptop though. I think you would be at a disadvantage if you decided to take your exams by hand. It's just so much easier to organize and edit your essays using a computer. However, there are ways to rent/borrow laptops for exam period, you don't necessarily have to own one.
With that said, you'd be ok with a desktop only, but I strongly recommend having a laptop. It really makes life easier for me - I can get work done on my computer no matter where I am as there's wireless everywhere these days. Laptops are also getting smaller and cheaper, which is great for law students.