To change the start date and/or due date of a bug, click on the current date or the - icon if one has not yet been set. This will expand a calendar pop up where you can click on the appropriate date to set.
Users with view and edit permissions in bugs can edit the dates. Depending on the settings of the user's project permissions, they may be able to view all the project bugs or just a subset of them. Watchers specifically added to the bug will be able to view it.