Can and impingement in the shoulder go away on its own?

Theoretically yes: Shoulder impingement is usually related to combination of inflammation and weakness with or without a mechanical irritant (e.g. Bone spur/ hooked acromion). With relative rest (avoiding aggravating activities) an impingement without a mechanical irritant can improve. Non-steroidal anti-inflammatory meds and rotator cuff specific exercises usually help. Not better may need surgical intervention.

Yes: The symptoms associated with impingement can. While the subacromial space doesn't get better the symptoms associated with the process may fade. Transient irritation in the bursa or rotator cuff in the form if bursitis or tendinitis may get better, of course treatment whether it is simple strengthening of the other cuff muscles. Or NSAID or injections may help the process go faster.