Yes the radiation from the infalling body gives the remote observer some information about what is going on with the body. From their viewpoint, the process of

Yes the radiation from the infalling body gives the remote observer some information about what is going on with the body. From their viewpoint, the process of fall into the hole takes infinite time, or in other words, it never completes. The longer the time of observation, the longer the wavelengths of the radiation from the body, so the less and less information is obtained, until the radiation is so red shifted that it is not measurable. That is when the acquisition of information stops. The body still falls down after that, it is a never-ending process but for practical purposes the body is gone.