Some blending modes can not be achieved with ink on paper (CMYK). Screen is one such mode (as is Overlay, dodge, etc) You can't "lighten" an ink by placing another ink on top of it.
If your Illustrator Document Color Mode is CMYK, this failure in the Screen blending mode should have been visible the moment you set the mode to Screen. You must be working in RGB. If you are outputting to CMYK, it's best to work in CMYK. Primarily so you get a closer representation of what any output may look like.
To fix your issue, you can try Flattening transparency in the file. (Object > Flatten Transparency). This may or may not work well for your particular artwork.
The other option is to restructure the art so that you do not use the Screen blending mode.
Since the file must be in RGB color mode, you could also try copy/pasting the art into Photoshop, or placing it into Photoshop as a smart object, Convert to CMYK there and Photoshop should do a decent job of converting the existing RGB artwork to CMYK. Photoshop sort of pseudo-flattens transparency upon conversion to CMYK. (Photoshop has an underlying RIP to process stuff like this. Illustrator does not.) This will result in a raster image rather than vector, but if the PPI is correct that may not make any real difference.
