I saved a map layer created by a another user. It is now sitting in "My Content". When I save that map layer, is it dependent on the original user's layer? In other words, if the original user deletes their map layer, will my layer I saved stop functioning? Or are both layers now independent from each other?
The saved layer is referencing the original data source (one way), i.e.,
If you change the configuration of your copy of the layer, this does not affect the configuration of the source layer. If you delete your copy, the original data still exists in the source layer, but you lose your layer configurations. If the owner of the source layer deletes it or removes access (in other words, they stop sharing it in a way that allows you to access it), the data is no longer available in your copy and will not display in your map.
To create a copy of the data from a hosted feature layer that does not maintain a reference to the source hosted feature layer, export the data from the hosted feature layer, download the file containing the data, and upload and publish that file as an independent hosted feature layer. To use this workflow, the source hosted feature layer must be configured (by the owner) to allow others to export from it, and you must have privileges to publish hosted feature layers.
For more information, please check Copy and save layers—ArcGIS Online Help | Documentation.