To determine the distributions it is sufficient to have two vectors in $S^1$, with the first coordinate not having the same absolute value, and the assumptions of the Hamburger moment problem.
Claim 1: Suppose $(v_1,v_2), (w_1,w_2) \in S^1$ with $v_1^2 \neq w_1^2$. Then $v_1^k w_2^k - v_2^k w_1^k \neq 0$ for any $k \in \mathbb{N}$.
Proof: Assume $v_1^k w_2^k = v_2^k w_1^k$ for some $k \in \mathbb{N}$ and lead this to a contradiction. If $k$ is even, we can take the $k/2$-th root to obtain $v_1^2 w_2^2 = v_2^2 w_1^2$. Now using that we are on the sphere, we get $v_1^2(1-w_1^2) = (1-v_1^2)w_1^2$ and this is equivalent to $v_1^2 = w_1^2$ wich is a contradiction to the assumption. If $k$ is odd, then the assumption holds only if $v_1 w_2 = v_2 w_1$ which is a special case of the even case. q.e.d.
Claim 2: Suppose $(v_1,v_2), (w_1,w_2) \in S^1$ with $v_1^2 \neq w_1^2$. Let $X_1, X_2 \in \mathbb{R}$, independent, satisfy the assumptions of the Hamburger moment problem. Then the distributions of $v_1 X_1 + v_2 X_2$ and $w_1 X_1 + w_2 X_2$ determine the distributions of $X_1$ and $X_2$.
Proof: We want to deduce the distributions of the moments of $X_1$ and $X_2$ iteratively. This suffices since both distributions satify the assumptions of the Hamburger moment problem. Suppose we know the moments from order $1$ to order $k-1$ for some $k \in \mathbb{N}$. By assumption we know the distribution of
$$ Y:= v_1 X_1 + v_2 X_2 \, , \quad Z:= w_1 X_1 + w_2 X_2 \, .$$
Consider $\mathbb{E}[Y^k]$ and $\mathbb{E}[Z^k]$:
\begin{align}
m_{Y,k} &= \mathbf{E}(Y^k) = v_1^k \mathbf{E}(X_1^k) + \sum_{l=1}^{k-1} {k\choose l} v_1^l v_2^{k-l} \mathbf{E}(X_1^{l}) \mathbf{E}(X_2^{k-l}) + v_2^k \mathbf{E}(X_2^k) \, ,\\
m_{Z,k} &= \mathbf{E}(Z^k) = w_1^k \mathbf{E}(X_1^k) + \sum_{l=1}^{k-1} {k\choose l} w_1^l w_2^{k-l} \mathbf{E}(X_1^{l}) \mathbf{E}(X_2^{k-l}) + w_2^k \mathbf{E}(X_2^k) \, ,
\end{align}
Using Claim 1 we can eliminate the $\mathbf{E}(X_1^k)$ term in a linear combination of the two previous lines and deduce an expression for $\mathbf{E}(X_2^k)$ only depending on $m_{Y,k}, m_{Z,k}$ and moments of $X_1$ and $X_2$ of exponents less than $k$. Thereafter we can calculate $\mathbf{E}(X_1^k)$. This provides the moments of order $k$ and we can continue iteratively. q.e.d.
For necessity, two points with $v_1^2 = w_1^2$ is not enough:
Thanks to Zeno44 for the good counterexample he provided; it extends to other cases with $v_1^2 = w_1^2$.
It is not clear to me what is the intuitive reason for the condition $v_1^2 \neq w_1^2$. The above "necessity" part is not the whole truth, since there is still the moment problem assumption. For this I can only provide the following comment.
Regarding the moment problem: Example 6.4 in an article by Svante Janson, http://arxiv.org/abs/math/0605642, tells you that it might be difficult to have the result holding without moment assumptions. However, he has no independence assumptions, so maybe one cannot easily find a counterexample in your case.
