If it is a cotton scrim, then use dye not paint - paint is a dye with a binding agent, and the binding agent will block the holes in the scrim, making it loose it's transparency, where as dye will just be absorbed by the cotton, colouring the fibres.
If it is a synthetic scrim, dilute acryllic paint at about 1:4 and test on a small square to make sure it is still transparent. If not, dilute further. If you dilute too much, the paint will not stick to the synthetic, and if you don't dilute enough, the binding agent will block the holes.
You can get some dyes that wash out - I have never tried them though, so I cannot comment on how good they are.
I cannot remember the exact proceedure for painting a scrim - but from memory, it is usually to prepare a full sized drawing of the design, lay the scrim over it and stretch it downwards slightly - to compensate for the load on the bottom when hung, then carefully paint - start from one side and work across, until finished. Allow to dry then carefully peel off the drawing behind it and hang. Set design is not my strong point, so hopefully someone else will be able to provide a better answer.