Is there a word to refer specifically to the people who live in the other part of a duplex (two-family house, side-by-side)? Not neighbor, i think, as that's more generic.
Following Kris's advice, I searched for a definition.
A duplex house plan is a multi-family home consisting of two separate units but built as a single dwelling. The two units are built either side-by-side, separated by a firewall, or they may be stacked. Duplex home plans are very popular in high-density areas such as busy cities or on more expensive waterfront properties. The two units of a duplex floor plan are usually a mirror image of one other, but are also available with attached units varying in size and layout. An example of this would be: unit A has two bedrooms and one bathroom and unit B has three bedrooms and two bathrooms.
For semi-detached houses which are side-by-side, the person in the other half is your next-door neighbour. For a maisonette where the building has two single-storey flats on different floors, that person is your upstairs (or downstairs) neighbour.
Next-door neighbour does include the properties on both sides of yours, so as well as referring to the person in the other half of your building, it also refers to the person on the other side, whose house yours is not joined to. In the UK, there isn't a standard term distinguishing those, although "through the wall" might be conveniently used to refer to the neighbour you share the building with.
In a terraced house, where more than one house is joined together along a street, even "through the wall" isn't particularly helpful, as that really only works for the end-of-terrace house (which are often sold as "semi-detached").
I don't think there is a distinction between separate houses and attached units. When I lived in a house, the people on the next property were my next-door neighbors. I now live in an apartment building. The person in the unit next to mine is my next-door neighbor, the person in the unit beneath is my downstairs neighbor. (Of course, we hope they don't call us the "people who stomp around on our ceiling".) As far as co-tenant, I don't think that would be accurate, as that (to me) indicates someone sharing the exact same unit, not the one next door. The two separate units in a duplex can be the property of two separate owners and so not be property held jointly.
I don't believe there is a well-understood special word for the people occupying the other half of the duplex you live in. Probably "co-tenant" or something similar is technically correct, but I imagine you would be misunderstood by many people if you tried to rely on that word alone.
In casual contexts I'd use something like duplex neighbour. I think almost everyone would understand that correctly.
In a formal context, like a police report, I would spell it out with a long phrase -- the occupant of the adjoining duplex apartment or something like that.