Given two Borel measures $\mu_1$ and $\mu_2$ on $\mathbb R$, is there always a Borel measure $\mu$ on $\mathbb R$ such that
$$ d\mu_1=w_1 d\mu,\qquad d\mu_2=w_2 d\mu, $$ for some functions $w_1$ and $w_2$ on $\mathbb R$ ?
Yes, if both measures are $\sigma$-finite. Then for such measure to exist it should be s.t. $\mu_i$ are absolute continuous w.r.t. $\mu. $ Just take $$ \mu = \mu_1+\mu_2. $$
If at least one of the measures is not $\sigma$-finite then as @GEdgar mentioned, there is a counterexample. Take $\mu_1$ to be Lebesgue measure and $\mu_2$ to be counting measure. Assume that measure $\mu$ exists, then