[logandavis]

Actually, there are "junior product" positions at many large tech companies. They're called "APM programs" (the A is for associate) and tend to hire new graduates with engineering backgrounds. Hires are rotated through multiple projects and usually given a lot of mentorship. I believe the first such program was created by Marissa Mayer at Google, and other companies later emulated it in an attempt to recapture its success. The current programs I know of are:

Google | Facebook | Uber | Yahoo | Twitter | Yelp | LinkedIn | IBM

There may be others that I haven't heard of, so feel free to respond with more. Google and Facebook are the big two; the others only accept single-digit candidates per year. Many of these programs are extremely new. (In fact, Twitter's is so new that it has no graduates yet, as far as I know).

Source: I'm a college senior with an engineering degree. I have applied to many of these positions.