Drop shadows and Bevel effects are Resolution Independent Effects (RIE). What ultimately dictates the resolution of these objects are the devices that renders or rasterizes the object. While object is in Indesign, the drop shadow or bevel effect is limited to monitor resolution. If you flattened this object, you will rasterize the effect based on the flattener setting selected and of course remove the resolution independence aspect.
Unfortunately, it's not just with how drop shadows and other effects view on the monitor. Some people are experiencing preflight flags on drop shadows. Even after making a hi res PDF. I didn't for a long time until just recently. For some reason, the same graphic that exported drop shadows just fine once before, is now being flagged by Indesign's preflight as low res AND is being created in hi res PDFs as low res.
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For some reason, the same graphic that exported drop shadows just fine once before, is now being flagged by Indesign's preflight as low res AND is being created in hi res PDFs as low res.
If you are using CC2018 make sure you are updated to the latest 13.1. There was a problem with images exporting as low res with the initial release and it has been fixed with the latest version. You might have to reset your prefs after the upgrade.
Shadow effects do get exported as an image and the resolution will depend on the size and gradation of the shadow. If you preflight in Acrobat, drop shadows can trigger a resolution rule violation because they don't have much to resolve (see #27 in my link above).
So if I preflight this PDF/X-4 with a simple circle fill with a long drop shadow, it will violate the 250ppi threshold. For print output the 175ppi is more than enough to resolve the shadow's limited number of gray levels.
On screen you have to consider the problem of the zoom level performance and resampling for the preview. If you zoom in and out the quality of the shadow is going to change, but that doesn't affect the print output quality.