You cannot use two different media for co-culture experiment. You will have to get both the cell lines acclimatize to a single media either DMEM or RPMI. If this does not work, you could try a totally different media in which both the cell types show optimum growth.
So, you will have to initially do some experiments to choose an ideal growth media suitable for both cell types. Check the health status of the cells in terms of viability and morphology in the common growth media you finally decide to choose, and then you could proceed with your co-culture experiment.
When cells grow in DMEM, they also will grow in RPMI, there is more stuff inside. Look at the composition. RPMI is more expensive, so people use DMEM when it's sufficient.
Culture your DMEM cells in DMEM-RPMI mixtures and finally RPMI for some passages before your experiment, to avoid strange results due to abrupt media change.