Actually there are two switches to flick in order to enable server collaboration. There is a global one that enables/disables the creation of server objects for the entire system (QMC->System->Setup->QlikView Server->Documents->Allow Server Objects) and there is one additional switch-per-document that will only function if you enable the global one. See QMC->Documents->User Documents->Select a document->Server->Server Objects->Authorization->Allow Creation of Server objects. The latter also allows you to grant permission to all or only specific users.
If you have a Publisher, lookup the corresponding setting in each task definition, as a task will overwrite all User Documents settings with those defined in the task configuration.
In QMC, there is no separate configurable permission to allow specific users to move existing objects in an AccessPoint document. That ability is simply governed by means of the Move/Size checkbox in the Layout tab of your objects properties. Even if Server Objects have been disabled (globally or just for this document), enabling the Move/Size object checkbox will allow everyone to toss around those objects in the AP.
Remove Last Document State will restore the document to its former glory. Very useful whenever someone turned your document into a mess.