We too have seen both approaches and are even familiar with some firms who, as part of their advertising, tout leaving the existing shakes and installing the new roof over them. They claim that by doing it this way, the job will go faster and will result in cost savings to the homeowner.
Baloney! We strongly recommend that all of the existing roofing be removed before a new roof is installed. The wood sheathing below the roof is highly susceptible to fungus damage and dry rot that will continue to spread if left untreated. Roof replacement is as excellent time to make a thorough inspection of the roof sheathing and make any needed repairs. This would be impossible if the new roof is installed over the existing roofing.
True, there is a cost attached to this, but, comparatively speaking, it is only a small part of what the new roofing job will cost. Besides, if dry rot does exist and it is not dealt with now, it could cost significantly more to make the repairs later.