I can’t speak for others, but as for my own research, at least half of my papers are joint with one or more authors, and amongst those papers that I consider among my best work, they are virtually all joint.
Of course, each mathematician has his or her own unique research style, and this diversity is a very healthy thing for mathematics as a whole. But I think 21st century mathematics differs from 19th and early 20th century mathematics in at least two important respects. Firstly, the advent of modern communication technologies, most notably the internet, has made it significantly easier to collaborate with other mathematicians who are not at the same physical location. (Most of my collaborations, for instance, would be non-existent, or at least significantly less productive, without the internet.) One can imagine the next generation of technologies having an even stronger impact in this direction (with this project possibly being an example; other extant examples include Wikipedia and the Online Encyclopedia of Integer Sequences).
Secondly, the main focus of mathematical activity has shifted significantly towards interdisciplinary work spanning several fields of mathematics, as opposed to specialist work which requires deep knowledge of just one field of mathematics, and for such problems it is more advantageous to have more than one mathematician working on the problem. (Admittedly, much of 19th century mathematics was similarly interdisciplinary, but mathematics had a much smaller diameter back then, and it was possible for a good mathematician to master the state of the art in several subfields simultaneously. This is significantly more difficult to do nowadays.)
The largest collaboration I have been in to date has involved five people – but already the dynamics of research change dramatically at that scale (especially when all five people are in the same room at once). One can toss an idea out there and have it debated by two other collaborators, while a fourth makes comments and corrections from the sidelines, and a fifth takes notes. Connections are made much faster, errors are detected quicker, and thoughts are clarified much more efficiently (often, I find one of my collaborators acting as a “translator” to distill an excited inspiration of another). It may not be “magic”, but it is certainly productive, and actually quite a lot of fun.