On open lists, you are only able to change the priority of items that you have created. This is why you may see a Read only badge next to the priority heading when creating/editing your items.
When creating an item (see Create an item) you can set its priority by choosing from the following options:
To choose an item's priority
1. Go to the list for which you would like to update the item priority.
2. Create an item or open the item you would like to edit.
3. Tap on the current priority underneath the Priority heading (the default will be Not set).
4. Choose a new priority from the popover and press Done to confirm your selection.
5. To save your changes make sure to tap ✓ in the top right of your screen. Your item will have the new priority and this will be reflected when viewing the item in a list of managing the item.