While you can share both public and private quizzes on Quizizz with your colleagues (please refer this link to know how), you still do not have the permission to edit someone else's public quiz. However, you can create a duplicate of that quiz and edit that copy freely.
Just go to the quiz you want to edit, and click the 'Duplicate' button (see screenshot below).
After the quiz has been duplicated, you will see 2 new buttons, "Edit" and "Delete" (see screenshot below)
You can now freely edit this quiz, and you will always find it again in your "My Quizizz" section.
Note: That this is your copy of the original quiz, and changes you make will only affect your copy, not the original quiz.