Indeed, there is huge respect for burial grounds in Middle-earth, in most of the cultures. I think that this sense of respect might in fact have led to feelings of superstition, even suspicion in cultures further down the line, and in other less respectful cultures, so this may have contributed to why these people thought the Downs would be a good place to hide. Even Orcs have their beliefs which would lead to superstition and an unwillingness not only to search the Downs for people but to even consider them as a likely hiding place!
Barrows (back in the real world now) were built over many thousands of years and by various cultures (as indeed they were in Middle-earth, as the Rohirrim still used them), by Iron Age and older cultures, but also by the Saxons and Vikings. So they had a lot of differences; there is also evidence that some very old ones could have bee re-used thousands of years later. Anyway...there are Barrows that seemed to be simple burials underneath mounds of earth, others that had openings which were sealed at a later date, and still more that remained open always - open in the sense that the way in could be seen, though it would be closed with stones or earth. Some 'Barrows' were not just burial places but also were used for other purposes.
Most of the looting would have taken place in later years sadly, by antiquarians, on whom the old tales of demons, witches and spooks would have no effect in scaring them off! Not all of them were as careful as archaeologists are today.
As for animals eating the flesh of the deceased, one of the old ways of burial was excarnation - whereby the corpse would be laid out where the elements might eat away the flesh from the bones; in some cases consumption by animals was avoided by enclosing bodies (like the Zoroastrian Towers of Silence in Iran and Tibetan Sky Burials - though birds would be allowed access), but in other cultures and times (e.g. in Britain in some periods), consumption by animals would be viewed as proper. I believe some saints' bones were also preserved.
Anyway I digress now I'm onto my hobbyhorse...the main point I suppose is that burial practices of other cultures can often be quite stomach churning which leads inevitably to suspicion and superstition, and that may have been why these Men took the chance to hide out in the Downs, which may have been a comforting place to them, being the 'home' to their ancestors.