You may be interested in this paper, which takes a semi-empirical approach to come up with a closed-form approximate formula:

• Jarai, Neda: On the size-distribution of Poisson Voronoi cells https://arxiv.org/abs/cond-mat/0406116

At the time when it was written, Mathematica did not have VoronoiMesh, but one could "trick" it to quickly compute a Voronoi tessellation by using ListDensityPlot with InterpolationOrder -> 0 and extracting the polygons.

Thanks for sharing everybody! Maybe this can add some more information:

In the article ["Statistical Distributions of Poisson Voronoi Cells in Two and Three Dimensions"][1] by Masaharu Tanemura, mentioned by Marco Thiel, the distribution of the perimeter of the cells (and the number of edges etc.) is also analyzed. Here I adapted the code of Vitaly to perimeters and we see (as in the article) that now we have a normal distribution:

perimeters = Perimeter /@ vorInner;
hist = Histogram[perimeters, Automatic, "PDF", 
   PlotTheme -> "Detailed"];
dist = FindDistribution[perimeters]
Show[hist, Plot[PDF[dist, x], {x, 0, .1}]]

And for the number of edges, we find a BinomialDistribution with FindDistribution although the result look almost "normal" as we use EstimatedDitribution

numberOfEdges[poly_th[First@poly]
edges = numberOfEdges /@ vorInner;
histE = Histogram[edges, Automatic, "PDF", PlotTheme -> "Detailed"];
distE = EstimatedDistribution[edges, 
  NormalDistribution[\[Mu], \[Sigma]]]
Show[histE, Plot[PDF[distE, x], {x, 0, 12}]]
distEB = FindDistribution[edges]
Plot[CDF[distEB, x], {x, 0, 12}, Filling -> Bottom]

I don't find it all that surprising that the result is a Gamma distribution. The 1-d case is well known as it is the distribution of interarrival times of a Poisson process and is exponentially distributed. The exponential distribution is a special case of the Gamma so maybe you're onto something.

Thanks! Awesome idea about edge-effects! Did you estimate any analytical models beyond Gamma or it was just pure empirical thing?