The power of randomized logspace with two-way access to the random tape
Well, here are a couple of observations. There's a famous PRG by Nisan that fools $\mathsf{BPL}$-type algorithms with seed length $O(\log^2 n)$. Given two-way access to the seed, Nisan's PRG can be computed in space $O(\log n)$. Therefore, every language in $\mathsf{BPL}$ can be decided by a $\mathsf{BP}^*\mathsf{L}$-type algorithm that only uses $O(\log^2 n)$ random bits.

In fact, you can get the best of both worlds. By using the same technique that Nisan used to show $\mathsf{RL} \subseteq \mathsf{SC}$, one can show that every language in $\mathsf{BPL}$ can be decided by a $\mathsf{ZP}^*\mathsf{L}$-type algorithm that only uses $O(\log^2 n)$ random bits.

So in terms of the state of the art, two-way randomness is more powerful both qualitatively (two-sided error vs. zero-sided error) and quantitatively (polynomially many random bits vs. polylogarithmically many random bits). However, it seems likely that $\mathsf{L} = \mathsf{BP}^*\mathsf{L}$ and hence the two models are actually equivalent.