I think the true answer is it depends.
Have you tried running any hardware testing software outside of windows or Ubuntu? This will help you determine if the bad sectors are hard or soft bad sectors.
-If they are "soft" bad sectors, chkdsk has already repaired them and both windows and Ubuntu will both write to those portions of the disk again.
-If they are "hard" bad sectors, you must start asking yourself things like...how old is my disk? have I caused any damage to the disk (like drops or falls)? If not, then there may be something wrong with the S.M.A.R.T controller on your disk. This piece of hardware is supposed to take care of marking bad sectors so that NO operating system running on the disk writes or reads to those areas.
-Preferably before but you can after; you perform your hardware test outside of all operating systems using a utility like pc-check you should make a backup just in case your drive is on the verge of failure. Hope this points you in the right direction.