I'd do as much as pushing as possible with the dull side and some pressure rather than the sharp side. That's just asking for trouble. Maybe for the gristle on the back of a coon's neck but that's about it.
And a good fleshing beam is just as important as the knife.

So my question Is for those who lay there hides on top of the beam. How does that work out when using the sharp side?

It takes more pressure to make a not so sharp knife cut the gristle that is very difficult if not impossible to push with the dull side. To me, the extra pressure needed to cut the gristle can result in cutting too deep. Tilting the blade angle up to get a not so sharp knife to cut also results in making cuts too deep also.

I want my knife sharp enough that I do not need to tilt the blade or apply hardly any down pressure to get the knife to shave the gristle.

I agree with other posts about the beam being smooth and the fur clean and free of debris.

But it is my opinion that poor technique is responsible for far more pelt damage than a knife that is supposedly too sharp. I have never personally had a fleshing knife that was too sharp but have struggled with ones that were not sharp enough to shave with proper angle and pressure. YMMV

I'm going to make a controversial statement here, if the knife is sharp enough to cut you easily its too sharp. Or more importantly, the angle is too fine.

A scissors is made to cut paper, try cutting your finger over one blade. We generally try to sharpen our skinning knives at 15 to 20 degrees, scissors at 40 45.

My fleshing knives are 40 degree bevel or more. They are super sharp and I don't cut holes in pelts. I only use the sharp side on coon and skunk necks and beaver backs and especially towards the tail. Everything else you push when on a beam. Make sure your beam is smooth and your fur is brushed of all burrs, mud etc...
Technique is everything and comes with time in the shed

When I first got my Post knife (arguably one of the best) it came razor sharp. How do I know? I took a rag and lightly cleaned it and it sliced it to shreds. The tips of my fingers also sliced a little when I just had to touch it because it couldn’t be that sharp.

A dull knife is way more work than a sharp one. The hard to push areas on a coon are a breeze with the sharp side. A slicing motion is used and hardly any pressure is needed.