During studies, most exchange students are staying in Academic hostel, which is situated just next to the university. 

Accommodation websites

However, some students are searching flats for themselves. Some websites might be helpful like:

On Facebook

Also you can find accomodation or flatemates within these Facebook groups:

More from TUT

In addition, have a look at the website TUT has put together on the subject: www.ttu.ee/.../accommodation