Does a second layer of OSB over a rotted layer cause weight or rot concerns?
Dryrot, unlike bug infestation, will stop when moisture stops. Plywood weighs 3.0 lbs. per square foot per inch of thickness. (OSB board could weigh slightly more because it has fewer voids and significantly more glue.)

So, adding a layer of 5/8" OSB board might add 2.0 lbs. per square foot at most...not significant, unless you live in a high snow zone or your roof structure has decayed too. (By the way, the Building Code allows up to 2 re-roofs before they all need to be torn off, which weigh about 4.0 - 6.0 per square foot depending on the type of shingle.)

A bigger issue for me would be what thickness of OSB board did they use? All roofing manufacturers require a minimum of 7/16". Less than that and you have no warranty...where the dryrot is located. 

Also, do the nails penetrate both layers of roof sheathing or extend a minimum of 3/4" into solid wood? All this info is on each shingle wrapper. 

In most jurisdictions, no Building Permit is required for re-roofing, (it's considered maintenance...that's why you don't need a permit for carpeting, painting, etc.) However, now they've replaced the roof sheathing (a structural item) and I'll bet they did not get a permit. So, who inspected the nailing of the new OSB roof sheathing? Big problem.