Lets nail down some figures with a random choice of compatible equipment with specs publicly available.
Gun Weight with empty magazine 670 g | 23.63 oz
Barrel Length 102 mm | 4.02 inch (slightly less than ammo test, so time under acceleration is slightly less)
Bullet mass 8.04 g (124 gr) Federal FMJ
Bullet velocity 1,150 ft/s (350 m/s) @ 118mm (4.65") barrel length, 0.00067s
Bullet energy 364 ft⋅lbf (494 J)
We could set this up as a conservation of energy problem, and that would be a reasonable approximation with a satisfying answer.
The time under acceleration is very short(0.00067s). (anything less than 0.01s feels instantaneous)
The gun weighs a lot (83 times) more than the bullet.
It is coupled to a human that weighs very much more.(180lb = 10155 times)
It is coupled over a large, soft, energy absorbing surface(hand).
The recoil is transmitted through several relatively massive energy absorbing tendons, tensed muscles, bones, flexed joints, fat, fluids, which redirect the recoil into countless vectors.
Where is the point of impact between gun and bullet? How does the gun impart force to the bullet and the bullet impart equal and opposite force to the gun? Well that doesn't really happen. It isn't a collision.
Only the timing is similar to a collision because both gun and bullet are accelerated in opposite directions by expanding gasses with same T0.
But is energy imparted equally? The center of those gasses is itself being accelerated away from the gun. Barrel friction counteracts recoil. You may be mistaken in part of your premise "momentum on both sides is to be equal".