Will steaming fish cause salt to come out or cause fish to fry in its own fat?

Salt moves due to differences in concentration, from regions of high concentration to regions of low concentration. So, when you boil brined fish, you surround the salty flesh with low salt concentration liquid, and thus the salt moves out to equalize the concentrations.

Steaming the fish will not result in as much movement of salt, because steaming applies less total liquid to the surface of the fish. Steaming works because the very hot steam condenses onto the cooler food, and by condensing it liberates heat (which transfers to the food). However, you still will have some movement of salt because the condensed steam on the surface of the fish remains at a lower salt concentration.

Also, steaming is much gentler on the fish, since the meat is not being constantly agitated by boiling water. With less agitation comes less leaching.

Grilling is the transfer of heat from pan to food, or from direct heat source to food. The temperatures are typically much higher than for steaming (steam is at 212F, 100C), so you get entirely different chemical effects from the process (e.g., caramelization). Therefore, I doubt the fish would "fry" in its own oil in the case of steaming.

If you're concerned about maintaining the salinity of your brined fish, you could consider steaming it en Papillote, or even in foil. This would result in the fish cooking in its own moisture, essentially eliminating the leaching of salt (although some salt would move to the surface, carried by the water).