It depends on who is responsible for the particular statue. Each one of these public statues has a public body (and it's going to be different) that is in charge of the care and maintenance of the statue. If the public wants a statue removed, they should file a petition with that organization, and there will likely be a public hearing. At the public hearing, the artistic significance, cost of upkeep, estimated value of the statue, cultural significance, cultural offense, speeches from representatives within the public, the art community, and all of it can be heard. Someone, in each of these cases, is in charge of whether or not that statue stays on the grounds.
That's generally the process. Doesn't seem like a bad one to me.
Also doesn't require defacing public property, trying to pull things down, or general panic or throwing up of arms.
Edit:
And to be clear, when I say "someone", it may end up being a panel.