Along with Kevin Garnett and Jason Terry, the 35-year-old made the switch to the Brooklyn Nets this off-season.
The Celtics were keen to rebuild their roster and sacrificed their ageing stars, but Rivers is disappointed Pierce in particular had to move.
"I didn't want Paul to go, even though I knew it was coming. That was a tough one for me," NBC Sports quotes him as saying.
"My thing is, Kobe [Bryant] is going to end up being a Laker for life. Dirk [Nowitzki] is going to be a Maverick.
"That's the one thing that, if we didn't do right, that was the one right thing we didn't do for Paul."
Pierce will instead now be hoping to help his new franchise reach their objective of winning an NBA championship.