Why does randomness have stronger effect on reductions than on algorithms? It is conjectured that randomness does not extend the power of polynomial time algorithms, that is, ${\bf P}={\bf BPP}$ is conjectured to hold. On the other hand, randomness seems to have a quite different effect on polynomial time reductions. By the well known result of Valiant and Vazirani, $SAT$ reduces to $USAT$ via randomized polynomial time reduction. It is not likely that the reduction could be derandomized, since it would yield ${\bf NP}={\bf UP}$, which is thought unlikely.

I wonder, what could be the reason for this asymmetric situation: derandomization seems quite possible in probabilistic polynomial time algorithms, but not in  probabilistic polynomial time reductions?
First, let me comment on the specific case of the Valiant-Vazirani reduction; this will, I hope, help clarify the general situation.

The Valiant-Vazirani reduction can be viewed/defined in several ways.  This reduction is "trying" to map a satisfiable Boolean formula $F$ to a uniquely-satisfiable $F'$, and an unsatisfiable $F$ to an unsatisfiable $F'$.  All output formulas are always obtained by further restricting $F$, so unsatisfiability is always preserved.  The reduction can be defined either as outputting a single $F'$, or as outputting a list of $F'_1, \ldots, F'_t$. In the latter case, "success" in the case $F \in SAT$ is defined as having at least one uniquely satisfiable $F'_i$ in the list.  Call these two variants the "singleton reduction" and "list-reduction" respectively (this is not standard terminology).

The first point it's important to note is that the success probability in the singleton reduction is quite small, namely $\Theta(1/n)$ where $n$ is the number of variables.  The difficulties in improving this success probability are explored in the paper

"Is Valiant-Vazirani's Isolation Probability Improvable?"  by Dell et al.

http://eccc.hpi-web.de/report/2011/151/#revision1

In the list-reduction, the success probability can be made large, $1 - 2^{-n}$ say, with a poly$(n)$-sized list. (One can simply repeat the singleton reduction many times, for example.)

Now, it is not at all evident or intuitive that we should be able to directly derandomize a reduction that only has success probability $1/n$.  Indeed, none of the hardness-vs-randomness results give hypotheses under which we can do so in this case.  It is much more plausible that the list-reduction can be derandomized (with a somewhat larger list).  Note though that this would not imply $NP = UP$: our output list of formulas may have many uniquely-satisfiable formulas, and perhaps some with many satisfying assignments, and it seems hopeless to try to define a uniquely-accepting computation over such a list.  

Even if we could somehow give a  list-reduction in which a satisfiable $F$ always induced a list $F'_1, \ldots, F'_t$ where most of the $F'_j$'s are uniquely satisfiable, there is no clear way to turn that into a deterministic singleton reduction for isolation.  The real underlying difficulty is that we don't know of any "approximate-majority operation for uniquely-satisfiable formulas", that is, a reduction $R(F'_1, \ldots, F'_t)$ whose output is uniquely satisfiable if most $F'_j$'s are uniquely satisfiable, and unsatisfiable if most $F'_j$'s are unsatisfiable.  This also seems like a general phenomenon: reductions output more complex objects than decision algorithms, and the properties of these objects are harder to check, so it's harder to combine many of these objects into a single object that inherits some property of the majority.

For the Valiant-Vazirani case, it does not even seem likely under plausible derandomization assumptions that we'd be able to obtain $NP = FewP$, that is, to deterministically reduce satisfiable formulas to satisfiable formulas with $\leq$ poly$(n)$ solutions.  Intuitively this stems from the fact that the isolating procedure has no idea of even the rough size of the solution set of the formula $F$ it is given.