I used to read this forum regularly when considering my own grad school journey, and thought it might be useful to share some perspective from the other side of the table: hiring for government affairs/public policy roles in the private sector.
I work for a large multinational corporation with sizable teams in these functions, and have been involved in the hiring process. With the caveat that every company is different (and this is solely my perspective), I have some general observations:
Work experience is king - Going directly from undergrad to one of these masters programs is a fool's errand. If you can get two years of work experience that's even tangentially related to your interests/goals, that's going to be seen more positively than a shiny grad degree.
"Prestige" is meaningless - In our most recent round of junior hiring (e.g. targeting people with 2-4 years experience), we received scores of applications from individuals with graduate degrees from Harvard, Yale, SAIS, and other universities generally seen as 'prestigious.' We ended up selecting someone with a master's from a state school, and someone without a grad degree (but several years of work experience). Having Princeton on your resume as opposed to U Texas won't do anything for you if your story and experience isn't compelling.
Employers want expertise - We want people who have at least some exposure and growing amount of expertise in our industry and set of issue areas. That's not to say that 24-28 year olds are expected to be true experts in a particular field--but having had related experiences is a major plus, especially if it's outside of the academic world.
But don't overestimate the value of regional knowledge - If you're applying to work for a multinational company in the U.S., odds are they have substantial offices and staff in various markets around the world. The added value of you speaking Arabic or Chinese, or having intensively studied these places, is limited.
Through a certain lens, these statements may seem discouraging. But on the contrary, I mean it to be liberating. Your life isn't over if you don't get accepted to Harvard or Georgetown or [insert prestigious school here]. And there are many employers in the private sector beyond the Beltway who see genuine value in smart young people who are interested in and have been exposed to public policy, government, and international affairs.