I have both a Bosch rotary hammer and a DeWalt rotary hammer drill. The Bosch instructions say to apply grease from time to time on the bits. Bosch even includes a small tube of grease.
DeWalt instructions never mention adding grease to the bit. When the DeWalt was factory new, a bit inserted came out with a light coating of grease.
So I asked at the (now former) local DeWalt service center. The tech there told me that I should continue to put a light coating of grease on the bits, particularly during times of heavy use or after times of relative non-use.
She also said that more than one Bosch rotary hammer user has failed to grease, and they notice that the chuck is a little dry when servicing the chuck.
The sense I had after the conversation was that one could not grease the Bosch, or grease the DeWalt, and unless extensive use was made of the unit, or extensive grease was applied, it wouldn't make a big difference.
All bits get warm during use, particularly hammer use. If there is an abundance of grease in the chuck, some may get warmed up and migrate out. Even without an abundance of grease, some may weep out.
I decided to grease my DeWalt (absent instructions from the manufacturer) and continue to grease the Bosch. I use a synthetic high quality chassis grease.