It depends on what type of feature service you are attempting to edit. From the Help:
Features are representations of real-world objects drawn on the map. Most maps contain a common set of features, such as polygons outlining
lakes, lines representing streets, and points showing specific places.
In addition to these common features, many maps use a focused set of
features to help you perform a specific task. For example, features
outlining recreation areas within a national park might highlight
potential conflicts between humans and natural resources, and a city's
evacuation routes, depicted as lines, can be a reference for emergency
aid organizations. Map authors can display attributes with pop-up
windows, change the symbology, and set the visibility range for the
features.
You add a set of features as a layer to your map which can either be
edited by you, the map author, or anybody viewing the map. Who can
edit features depends on the type of feature layer. Features from a
service layer are editable by anybody viewing the map, so they are
useful when you want community input. For example, you might include a
service layer that allows the birding community to post their bird
sightings directly on the map and attach media files, such as
photographs, audio files, and video files, to the specific observation
points. Because these features are part of the service, any changes
made on the map are immediately viewable to everybody who has access
to the service. This type of layer is useful when you want others to
edit your features, you have large amounts of data, and you have
access to ArcGIS for Server.
When feature layers are stored in the map, only the map author can
edit the features. In other words, your edits are saved when you save
your map. These types of feature layers are useful for displaying
information such as events happening within a community or inventory
of oil production facilities. If you have features in delimited text
file or GPX file, you can import them into your map. Feature layers
that are part of the map are also an easy way to add a small number of
features, for example, the swimming pools managed by your city's parks
and recreation department and when you don't have access to ArcGIS for
Server for creating feature service layers.
One way to get around this issue would be to transfer the ownership of the map to whoever is editing.
