Question: Who should I ask to proofread my MSc dissertation? Is proofreading required?

Answer: 
There's no formal process that you are required to use for proofreading, but it's certainly required in the sense that your dissertation should not be full of typos.

You are the most important proofreader. You care the most and have special expertise on the specific topic you are writing about, and eliminating errors is your responsibility. However, it's worth getting assistance from someone else, since some errors can be tough for the author to spot.

You should not expect your advisor to do proofreading. It's possible that he/she will provide a list of typos, but that's really not the advisor's responsibility. Ultimately you'll benefit far more from higher-level advice about content and writing, so it's in your interest to provide drafts that you have already checked carefully.

One good way to arrange proofreading is to swap dissertations with a friend, with each of you looking over the other's draft. If you can't find anyone to do this, you could try asking friends or relatives to help you proofread, but you'll have to be careful not to let it become a burden for them.

I would not recommend trying to hire a proofreader, unless you have clear approval from your university for exactly what you plan to do. (If I heard that a student had hired someone to help proofread, I would worry that "proofreading" might be a euphemism for an inappropriate level of writing assistance.)
I'm now reading the Msc dissertation of a friend but being brutally honest in my comments which she happens to like. It does help that I once was an archaeology student myself. But I think that the more people who read your dissertation before you hand it in, the better. The process of writing up a dissertation isn't just about writing it all up. It's also about being able to form up your own opinon, being critical, being able to work together with other folks and work on your own at the same time.