This is most probably caused by the sanitize setting of the widget. This will remove the images. Thjis can be turned off but has possible security implications, please read the documentation on this widget carefully.
Hi Ben Browne, to insert a image in rich text widget you have an options in tool bar settings of widget you need set to Full, or Standard where the option will be enabled to insert image, the same settings you need to keep at the other end as well.