Using iD Editor to for personal / alternative maps

For a kind of "My Maps" feature, you could try uMap. It lets you add markers and draw lines on top of an OpenStreetMap based map. You can then share that map with others, or embed it in a website etc.

I have a similar issue where I'm creating a new trail map for a Forest Service winter recreation area. There are a number of items to be displayed that are not appropriate for submitting to the OSM database.

I am using the JOSM editor and keeping these items saved in a local .osm file. I've setup some shell scripts to do things like generate UTM grid lines into a local .osm file, generate contours, hill shading, etc. into local shape files and then use a local custom Mapnik XML style to pull it all together into a map that can be printed. Probably "too much information", but the short answer is you can have local .osm files that you can edit and update using JOSM which you do not need to submit to the OSM database.

Whether or not you can do the same with a web based browser, I don't know. JOSM is actually fairly easy to use and I suggest you try it out.

With respect to having others help you on your local map: If you put your local .osm file into a distributed version control like git you can have more than one author working on it.

You did not add "simple" to the question, so, to be complete: Everything is open source here. You could copy the/a extract of the OSM database into your own database, set up and API and use iD with it. You can set up a renderer to create a map of your data. See for details this other question: can-i-make-a-private-osm-with-user-access-validation-and-custome-notations.

Okay, a few questions here, so I'll go one by one:

There's no 'My Maps' feature in OSM. It's one map for everyone, which is meant to be authoritative. There are plenty of sites that provide some level of personal features above OSM - uMap, MapBox, and so on.

Currently iD saves to OSM, and can be set to save to a different instance of the OSM software, but does not save to .osm files. It could eventually, but the use case is unclear. If you just want to draw points and lines and so on, the full OSM data model is overkill and will limit the tools you can use. edit-geojson would be a better option for that.

As far as 'collaborative side projects', this is more or less the first question: OSM hosts one map, for everyone, rather than personal maps. There are plenty of forked or privately-run copies of the OSM software for specific purposes, and they can interact with the 'main OSM' in a variety of ways, but it's usually a much less-optimal way to do things, since the merge and correspondence between the two is time-consuming.