When I was an intern we were given a choice between A) taking a 3k check and B) letting Msft find housing (which they then deducted about $600/month from your monthly salary to pay for)..
The first summer I interned they put probably 300 interns in a giant/new dorm/apartment on UW's campus - which was a lot of fun. The second summer I was in an apartment in downtown Bellevue, which was renting for about $2,500 at the time.
So numerically, you could A) earn (3k-your housing) or B) spend 1.8k and get Microsoft housing.
So basically if you wanted to break even (difference between the two options is 3k+1.8k=4.8k), you would need a 3 month lease with monthly payments of less than 1.5k total.
The major downsides I see with finding your own apartment:
A) it can be hard to find if you aren't in the Seattle area (no viewings, tours, etc)
B) finding a 3 month rental (VRBO type options will be expensive, but it will be hard to find apartment complexes giving 3 month leases :) )
C) you will have to find your own roommates, or go alone - it's nice to have someone interesting and different
D) mental stress
Basically I'd say if you can find a 3 month rental that will cost you less than 1.5-1.6k a month and doesn't cause too much other headache you could do it....
But let me ask you another question: do you know the neighborhoods of Seattle/the Eastside? Are you looking at living close to work (Redmond and Bellevue are pretty boring tbh, mostly older tech employees with families and young kids)? Do you want to be in Downtown or Cap Hill (most interesting but long commute, you could take the connector or city buses too)? You could go by U-District but be prepared for college-like housing. If you don't really have a clue it's probably for the best they find a roomie for you and house you somewhere in downtown Bellevue haha
source: 2x intern, starting fulltime in March, lived in UDistrict+Bellevue+Belltown(currently)