In the situation you describe there are two modes of vibration that can occur:

1. The springs "see saw" - with one rising while the other falls, as you suggest. We describe this as the two springs being 180 degrees out of phase.

2. The springs act in phase, so that the mass bobs up and down as the springs work in unison.

This is known as a 2-degree of freedom spring-mass system. The motion of the system is the linear combination of the two modes of motion, with the amplitude of each mode determined by the initial conditions of the system. To solve you would need to set up the ordinary differential equations governing each mass, and then solve to find frequencies of each, and finally determine the amplitude of each based on initial conditions. You can get an idea of how to set up the equations here:

http://www.efunda.com/formulae/vibrations/mdof_eom.cfm
Not knowing what level of mathematics you have attained - if you are familiar with matrix algebra and solving linear partial differential equations we can probably continue this discussion in some detail. If not - we will have to keep this pretty general. Please post back with any specific questions you may have on this.