There aren't enough of us that I can specify where I went without giving myself away, but I will say that my location in no way impacted my ability to get my job. Networks help and I think most of the schools outside of DC have networks in DC as strong as SAIS/Georgetown. I got my job through a combination of luck (applying online among the masses) and networking (met fellows before position opened up).
Salaries at Brookings are terrible. Having a MA maybe gets you another $2/3k-not nearly enough to make up for grad school loans. I should also add that prestige of MA matters hardly at all at Brookings. I have as many coworkers from one year overseas programs/lower ranked schools as I do people from the top programs.
Brookings is a fantastic place in terms of the research work the institution puts out and the fellows you can work with, but you really max out your responsibility level after a year. Most RAs stay 2-3 yrs, a few less/longer, but promotions are name only and growth opportunities practically nonexistent. I think it's a great place to work right after undergrad before going to grad school but less valuable afterwards. That being said, so many people want to work here that the norm does tend to be that people are hired after grad school and with several years of experience.
Upward mobility is practically nonexistent in the think tank world-at least, when you are hoping to go from research assistant to fellow. You definitely need either a PhD or a very prestigious government career behind you. Also, most internships are targeted towards undergraduates-grad school students end up in them just because lots of people wanting a place like Brookings on their resume means that people hiring interns can take grad students over undergrads, but the level of responsibility is extremely low. Also, jobs open up so scarcely that it's practically impossible to go straight from intern to staff. If you're looking for a career track, government is definitely the way to go.
I got my job through mostly luck-I applied online, and had met some Brookings fellows in the past who I was able to reach out to, which I think helped me make the first cut. Some people get jobs here without having met anyone at the company, others are former interns (likely people who interned a year or so before coming back), others are internal hires. If I could have done things differently, I would have preferred to have my job before spending time and money on grad school, because I think experience matters quite a bit in DC. I only had about a year, year and a half of full time work and a summer internship on my resume coming out of grad school, and I have coworkers now that are at my title level, don't have a MA, and are 2/3 years younger than me. It's tricky because I do think that many fellows prefer MA candidates when they are hiring, but I think the best way to get into a place like Brookings is to intern during undergrad/straight after, make a good impression, and get hired early as a researcher or coordinator. You get a lot more out of this job if you haven't gone to grad school yet, whereas people like me get stuck, because we can't move up to the fellow level and can't use grad school as a means of making a career switch.