Three approaches were debated during the Constitutional Convention -- election by Congress, selection by state legislatures and a popular election, though that was restricted to white landowning men.
No written law or rule requires the senators to remain silent on the issues. But it's probably a good idea, and a promising sign of fairness.
The Founders saw impeachment as a regular part of ensuring presidential accountability. A constitutional scholar offers a possible process for a rapid and smooth impeachment inquiry.