We have in our workflow a transition called "Accept" - from status "Open" to status "In progress".
When the agent "X" accept the request, the assignee of this request should be changed automatically to agent "X". This should work for all agents.
When the agent "X" use in a new created request with status "Open" the function "Assign to me", the status should be changed to "In progress".
Which triggers should I choose and which automations should I define to get this automatically functions?
It's not possible using only the automation available on Jira Service Desk, we use a plugin named Automation For Jira that can automate your workflow, it can be triggered when a Issue is created, transitioned or edited using JQL for checking, and then take action, like assign the issue.
For your first use case, just add a post function to the Accept transition of type "Assign to current user". That will cover case 1.
For the second case, Rafael's suggestion might work (there is also a free version of the add-on, Automation Lite for JIRA), but before you go down that path - ask yourself if you want that behaviour. Changing the assignee of an issue is a totally different action than executing a workflow transition.
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