The Senate waits only if the full House votes to approve an impeachment inquiry.  Pelosi didn't do that because she didn't have to votes.  That's why they're trying to back door it.  Because it isn't an official impeachment inquiry, the Senate can call the same witnesses that are called by the house and essentially do the same thing.  Why?  Because the House is acting in secret and can't be relied on to tell the truth about what is happening.  The Senate could conduct the same "unofficial" inquiry to keep it honest so the Dems can't twist and craft a narrative based on snippets of info without context.