Here it is with some comments:
# import necessary packages
using SparseArrays, Distributions, Random
# initialize a random number generator with a seed so that
# results are reproducible (not mandatory but good practice)
rng = Xoshiro(1)
# generate a Boolean "'mask" matrix to decide where the nonzeros are
M = sprand(rng, Bool, 10, 10, 0.5)
# allocate a matrix of the same size and nonzero structure
# but with a floating-point element type
N = similar(M, Float64)
# modify the non-zero coefficients of N in-place
# with the distribution of your choice
UD = Uniform(2, 3)
rand!(rng, UD, nonzeros(N));
# voila!
You can but it will be much less efficient in high dimension since you generate dense matrices before converting them to sparse format.
If you want the devs to be aware of that, the best way is to open an issue on the Distributions.jl repo.