The sentence "Both of us won a prize" does not have a double meaning to me. In my mind, the meaning is clear: You won a prize and I won a prize. You could also say "Both of us won prizes."
If you mean that we won a single prize together, I might say "Both of us won the prize," or "We were co-winners of a/the prize," or something else that tends to remove any ambiguity.
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