A general technique to check for ambiguity is this. Transform the grammar in an equation system:
$S(x) = S(x)^2 + 2x$
Solving for $S(x)$ yields $S(x) = \frac{1}{2}(1 - \sqrt{1 - 8x})$ which is the ordinary generating function enumerating the grammar's left-derivation trees.
The generated language is $\{0,1\}^+$ which has the generating function $\frac{1}{1-2x} - 1$. This is obviously different from the above, therefore there are more left-derivation trees (for words of length $n$) than words (of length $n$).
This technique goes back to Chomsky and Schützenberger, 1963 (PDF).