Had a new carpet fitted on the stairs. Full width.
They have fitted grippers in the bottom back of each stair, but none at the sides.

The sides of the carpet sort of float in the air a bit rather than staying tucked down.

The missus says there were side grippers there before. The fitter says there weren’t. I don’t think there were, but not sure.( and TBH, immaterial what was there before, just want it right now)

Should there be grippers at the side?
Should the sides remain tucked in?
Or should the ( almost) edge of the carpet just sort of be there?

Gripper rods are not needed and is not standard practice for the side edges, just the two bits in the crease of the stairs to keep it in place. This is hessian backed carpet i take it, the sides should be cut to neatly sit against the stair frame. When you say tucked in are the sides folded over?

Never seen them at the sides of stairs/steps. Only ever at the back of the step near the riser.

Have you a thicker underlay on the new carpet?

Ours has grippers at the back, not the sides. There’s enough tension in it that the sides stay flat and don’t lift.