I think you need at least one person who is completely independent of all others. For example if the same random is added up twice (say if the room is just you and your evil clone), you can get stuck with just even numbers. Independence is a pretty strong property to ask for since we all share the same biology...

But if you do have independence, the proof is easy! Let S be the sum of everyone else and X be the discrete uniform random in [0, 9]. Then:

  Pr(X + S mod 10 = i) 
    = \sum_j Pr((X + j) mod 10 = i | S = j) Pr(S = j)
    = \sum_j Pr(X = i | S = j) Pr(S = j)
    = \sum_j Pr(X = i) Pr(S = j)
    = Pr(X = i) \sum_j Pr(S = j)
    = Pr(X = i)

Due to total probability, symmetry, independence, and total probability again respectively. The handwaved part is the mod where you can imagine the histogram columns in the pmf getting rotated/shifted around but ending up looking exactly the same afterwards since all columns are symmetrical.