how to formulate query (conditional)
If I understand your question, you should think about it this way:

Select aat.action_id, aat.emp_id
    from ActionAssignedTo aat
    where aat.action_id = 10


gives you the keys you want.  You just expand that to include look-ups to the other tables to get the rest:

Select {whatever columns you want from any of the tables}
    from ActionAssignedTo aat
    inner join action a on a.id = aat.action_id
    inner join employee emp on emp.id = aat.emp_id
    where aat.action_id = 10


In your query, you are limiting the results to only action 10 in the second INNER JOIN and then further limiting the results to employee 1000 in the WHERE clause.  It's not employee 1000 and all the other employees who were also assigned action 10; it's just all the employees who were assigned action 10 (employee 1000 will be among them).  The solution (unless I'm totally misreading what you're trying to do) is fairly simple and your original approach was over-complicating things.

I hope this helps.

AMENDED:

I looked at your newly posted query.  Still looks like it;s getting in it's own way but it made me wonder if you're trying to test for Employee 1000 as either the assigner or the assignee (since you're joining the employee table to itself again).  So I added adding logic to do that additional test in the where clause on the off-chance that is the sticking point:

Select {whatever columns you want from any of the tables}
    from ActionAssignedTo aat
    inner join action a on a.id = aat.action_id
    inner join employee emp on emp.id = aat.emp_id
    where aat.action_id = 10
        and (aat.emp_Id = 1000 or a.emp_Id = 1000)


I'm inferring that ActionAssignedTo.emp_Id and action.emp_Id represent the assignee and assigner respectively - hope I got that right,

I hope this helps.