In order to do that you need to check on the workflow assigned to your project what are the Conditions set for your workflow transitions, by default JIRA as a condition set that says "Only the assignee of the issue can execute this transition." If you remove that condition or add more people to it you should be able to transition your users issues.
Check this page, it contains more information about workflow conditions: Advanced Workflow Configuration
Hope this helps!
Additionally you should check for a postfunction which sets the assignee to the current user. If you only wan't to change the status to "In Progress" you have to remove this postfunction.
Because this can lead to confusion maybe you should consider adding an additional transition from Open to In Progress for administrators only, something like "Start Progress (Admin)" which is only visible for admins (use a condition to restrict visibility).
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