There are permissions based on user role for access to reports. There are no user specific permissions built into Exports and Reports, however you could add an action with PHP to ensure a limitation by the current user ID is utilized.
The following code assumes that you have a “user” field defined in your report and filtering is enabled for it. It will always force the user filter value to be the current user ID and will override filter access.
add_action( 'wp_admin_ui_pre_export', 'my_force_user_filter' );
add_action( 'wp_admin_ui_manage', 'my_force_user_filter' );
// Handle filtering of user for reports page.
function my_force_user_filter() {
// Filter only when on the reports page.
if ( 'exports-reports' === $_GET['page'] ) {
// Force the user filter to always be set.
$_GET['filter_user'] = get_current_user_id();
}
}