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Yes, hosted tiled map services don't return any feature geometry, nor do they support any attribute queries against the data. It's just a tiled cache of images of the data.
Having said this, for your map services on your own GIS server, you can turn off the Data capability. This will prevent access to any geometry. However, it will also prevent queries against the data. Alternatively, when authoring a map in ArcMap to publish as a map service, if you hide the shape field for the layers, this will prevent the feature geometry from being accessed. In this case, attribute queries are still enabled. I have not tried this latter case myself...
Thanks,
Mike