This is controlled by the global value:
$g_handle_bug_threshold

The description of this from the config_defaults_inc.php file is:

access level needed to be able to be listed in the assign to field.

If you want to give reporters the ability to change the priority level, you would have to add the following to config_inc.php file:
$g_handle_bug_threshold = REPORTER;

This may have unwanted side-effects like allowing your reporters to assign bugs (to other reporters even).

Seems like maybe adding another variable to control access to priority may make more sense. Then again I am new to Mantis but I'm sure someone with more experience could clue us in on why changing priority is tied to being listed in the assign to dropdown.

The original intent was that a user sets the severity of the problem. The developer / management team sets the priority after review. After that, the priority would set the order of implementation. In a business a crash by a single individual in an odd configuration should not be more important than a major annoyance for all users.

I agree. Many users would believe their bugs to always be a top priority and thus would make that field ineffective.

In our case, the customer would set his view of priority. Then the manager can set another priority after a review. For us it is certainly an necessary to configure this field.
(ascom Switzerland)

I think there are several things to fix:

1. The documentation of $g_handle_bug_threshold in config_defaults_inc.php should report the dual use of this field (for assigned to and for priority)

2. As it is a typical setting that the reporter may have the handle (assigned to), this should be an option, even if we don´t want the reporter to set the priority.

3. As priority is set internally, there should be an option to make it "private".
   (the last requires some changes at least in:
   bug_report_advanced_page.php
   bug_report_page.php
   bug_view_advanced_page.php
   bug_view_inc.php
   bug_view_page.php
   my_view_inc.php
   config_defaults_inc.php
   core/custom_function_api.php )

This can be achieved in current version of MantisBT, by customizing the bug_report_page_fields config option.