The wedge should release with some firm thumb pressure on the rounded end. Newer Pietta's seem to have wedges fitted with far too much pressure, resulting in difficult initial removal requiring brass punches and hammers.
Someone may have "tinkered" with the wedge to make removal easier by thinning the sides and/or top. The rounded edge should protrude slightly, perhaps 0.010"
You can make some adjustments by very slightly bending the wedge: hold the flat rounded end in a vise, then tap the flanged end with a brass/plastic face hammer. I don't think it makes any difference which direction you bend the wedge.
This is a trial-and-error process: tap a little, try, put it back in the vise and try some more. You won't need much of a bend to restore the friction fit. If you can see the wedge bend when you hit it, you hit too hard!
Next choice is purchase and fit a new wedge.
Best of luck!