Speaking as one who was accused of bullying
Every case is different. Briefly, in my case, I felt it desparately unfair, I had tried hard to help the person concerned who had performance capability issues, and was utterly unaware that this was being taken as bullying. I still find it hard to understand. The complaint - the very first complaint - was made in the form of a legal letter. There was no attempt at any informal resolution. The result - I resigned immediately because I felt utterly overwhelmed and destroyed. The other person got a small settlement in the end - largely because companies really dread going to tribunals (time consuming, costly, and the outcome is uncertain) - and also left the company. In effect everyone lost.
Different place, different industry, different circumstances. But my point is this. Raise the matter with the person concerned, or if you understandably would rather not, then do so with HR. They will take it seriously. Companies are very concerned to nip this in the bud because the payouts can be very very high if they don't.
A formal complaint will lead to an investigation - which may be the only solution, but is likely to be pretty distressing for you and people around you (who may be questioned in the investigation). Assuming the complaint is upheld, disciplinary action for gross misconduct (and therefore potentially summary dismissal) is a likely outcome for the person accused of bullying. Do it if you have to, but do try to make it your last resort and see if you can resolve the problem less formally.