How to draw a common boundary in JOSM?

Let's say you have two adjacent areas a bit like the geometry formed by the letters "CD".

You would create three ways; one for the "C", one for the "D"'s belly, and one for the line that separates them (let's call it "I" because it looks like one). These ways would go untagged, or maybe tagged boundary=administrative so that people know what they are. Then you would create two relations, one containing the "C" way and the "I" way, and one containing the "I" and the "D" way. The relations should be tagged type=multipolygon, boundary=administrative, admin_level=x (whatever level it is), and name=y (each with its own name).

Some people also use type=boundary but multipolygon is more commonly understood.

The above is the "proper" method that can be used for any number of adjacent areas - if you have a mesh of areas then your relations will often have more than two members. In its relation editor, JOSM provides a visual clue as to whether all the ways you have assembled in that relation really form a closed ring or not.

There is also a simpler method that you can use if the "I" way is rather short and has few nodes only. In that case, you can simply draw both areas as a closed way - two ways only, no relations - and have them share the nodes of the "I" way. This is perfectly ok for small stretches of coincidence. It avoids the relations which are difficult for many newcomers, but at the same time modifications to the shared "I" border will often be more difficult.