Piercing the Corporate Veil

Can you explain “piercing the corporate veil”?

We recently bought an STR (with three other couples) and its undergoing a giant rehab.

Is it safe for each of us personally to promote it?

Is it okay for people to know who the owners are?

If it is a known fact that I’m an owner does that put me at risk if there were ever a lawsuit. Would the veil be pierced if it becomes common knowledge that we own it?