How do I mark a task completed for one assignee, but keep the issue open.

[Sean Haskins]

For our software development team, we have multiple steps in the process carried out by different people. A bug fix may be developed by one person, tested by another, and deployed by a third. We have the workflow defined in Jira. However, when John finishes development he can remove himself as an assignee to keep the task open, but his work won't be tracked when we review productivity at the end of the sprint. 

[Laurie Sciutti]

Hello @Sean Haskins and welcome to the Community!  This scenario is a typical use case for including sub-tasks in your Bugs/Stories.  Have a look here.

[Nic Brough -Adaptavist-]

Welcome to the Atlassian Community!

That is not how sprints work.  The point of a sprint is that the team selects some issues to deal with, and then delivers results.

It does not matter that several people might work on it, nor that which team member does any work on it.

You should be tracking team productivity, not individuals.  "task open" means that the team has not finished.  "task closed" means the team has.