Plot3D[x[s1, s2],
  {s2, (m3^2 + m^2)^2, (Sqrt[s] -  m^2)^2}, {s1,s1min[s2], s1max[s2]},
  RegionFunction -> Function[{x[s1, s2]}, -1 < x < 1]]

ClearAll[x, m, m3, s1min, s1max]
x[s1_, s2_] := Sin[s1 + s2];
m = Sqrt[Pi/32];
m3 = Sqrt[Pi/64];
s1min[s_] := Sqrt@s;
s1max[s_] := 2 + Sqrt@s;

Plot3D[ConditionalExpression[x[s1, s2],
   (m3^2 + m^2)^2 <= s2 <= (Sqrt[s1] - m^2)^2 &&
   s1min[s2] <= s1 <= s1max[s2] && -1 < x[s1, s2] < 1],
 {s1, 0, Pi}, {s2, 0, Pi}, PlotPoints -> 100, MaxRecursion -> 4]

Plot3D[x[s1, s2], {s1, 0, Pi}, {s2, 0, Pi}, PlotPoints -> 100, MaxRecursion -> 4,
 RegionFunction ->
  Function[{s1, s2, z}, (m3^2 + m^2)^2 <= s2 <= (Sqrt[s1] - m^2)^2 &&
    s1min[s2] <= s1 <= s1max[s2] && -1 < z < 1]]
 (* same picture *)