If the gun is strictly for self-defense, I'd say that the .357 Magnum is a better choice. While I'm not hesitant to use a .44 Magnum as a defensive revolver (a 4" 629 is my bedside gun more often than not) it does have certain limitations. The biggest problem, as I see it, with a .44 Magnum is that the selection of good self-defense ammo is pretty limited and simply stuffing any old .44 Magnum ammo into a defensive revolver is a very bad idea.
The problem with a .44 Magnum, ammo-wise, is that most of the ammo is made for hunting and, as such, will likely expand little, if at all, on its way through an erect biped. While .44 Magnum ammo isn't difficult to find, it's often a JSP or very tough "controlled expansion" JHP bullet which isn't really a good choice for self-defense in this caliber. While .44 Special ammunition can be use and that caliber can be quite a good defensive cartridge in its own right, the selection of ammunition is even more limited in .44 Special. When one can find .44 Spl ammo in stock (and you can't always) it's often loaded with SWC or LRN bullets.
The second issue with the .44 Magnum is its recoil. While it can be shot quickly and accurately, it's not a easy thing to do. While I can shoot my .44 Magnum revolver with my carry ammunition quickly and accurately enough to feel comfortable using them for self-defense, it takes a lot of work to get there and pretty regular shooting to maintain it. Also, I'm shooting an all-steel gun, I've put big, cushy grips which fit my hands well on it, and I'm not using the most powerful .44 Magnum ammunition I can.
Finally, the .44 Magnum is expensive in terms of both guns and ammunition. A good-quality .44 Magnum revolver will generally cost at least $100 more than a comparable revolver in .357 Magnum and it's difficult to find ammunition for less than $40/50 rounds unless you're willing to resort to mail order. Even if you handload (and you should if you plan to shoot much .44 Magnum), a .357 Magnum is still more economical as the bullet cost substantially less and you're not using as much powder for each cartridge.
A .357 Magnum, on the other hand, offers several advantages. In a medium or large frame revolver, its recoil is much more manageable. Just yesterday I was shooting full-power 158 gr .357 Magnums from my S&W M27 and the recoil was actually milder than standard pressure 158 gr .38 Specials from my S&W 442. The .357's recoil is much more conducive to developing proficiency and not nearly as punishing to practice with.
The .357 Magnum also offers a much wider variety of good SD ammo than the .44 Magnum does. While I have to search for a satisfactory .44 Magnum JHP, just about hollowpoint in the 125-158gr range will be satisfactory in .357 Magnum. Also, there are some very good options available in .38 Special too and they're much easier to find than .44 Special ammo.
Finally, while not as powerful as it's larger brother, the .357 Magnum is by no means a pipsqueak. It was a very popular cartridge for police before revolvers fell out of favor and enjoyed an excellent reputation for ending fights. If fact, when revolvers were the preferred cops' gun, most of the departments that chose not to use .357 Magnums did so because they felt it was too powerful rather than the other way around.