In the case of $Z=XY\sim \text{Beta}(\alpha,1)$, the moment generating function (mgf) of $-\ln(XY)=-\ln X-\ln Y$ is
\begin{align}
M_{-\ln(XY)}(t) 
   &= E(e^{-t\ln Z})
\\ &=E(Z^{-t})
\\ &=\int_0^1 z^{-t}\alpha z^{\alpha-1}dz
\\ &= \frac{1}{1-t/\alpha}
\end{align}
which is the mgf of an exponentially distributed random variable with rate parameter $\alpha$.  

Thus, if $X$ and $Y$ are iid, this means that $-\ln X$ and $-\ln Y$ must be Gamma distributed with shape parameter $1/2$ and rate parameter $\alpha$ (see [wikipedia][1]).
Backtransforming, the pdf of $X$ and $Y$ is
$$
f(x)=\sqrt{-\frac\alpha{\pi\ln x}}x^{\alpha-1}
$$
for $0\le x\le 1$.

A similar calculation in the general Beta$(\alpha,\beta)$-case leads to
$$
M_{-\ln(XY)}(t)=\frac{\Gamma(\alpha-t)\Gamma(\alpha+\beta)}{\Gamma(\alpha+\beta-t)\Gamma(\alpha)}
$$
and the mgf of $-\ln X$ and $-\ln Y$ would need to be the square root of this.  But that is not the mgf of any known distribution and such a distribution (that when convolved with itself produces the target density) may not even exist.

  [1]: https://en.wikipedia.org/wiki/Gamma_distribution#Summation