You must check your individual company's Standard Operating Procedures in your Operations Manual which should define the extent of responsibilities of each crew member.
Some companies allow FOs to reject take offs; with others the FO can raise a concern but the Captain has the ultimate decision whether to reject or not.
Generally speaking (and check that this does not contradict your own company's rules) you would stop for any malfunction (within reason) up to a nominated speed, eg 80kts, and then only stop for major problems up until V1, such as fire, control jam, runway incursion etc. One of the reasons for this is that a high speed rejection has implications for brake overheat and/or tyre bursts.
You will have a standard brief to perform prior to a right seat PF take off, and within this you should discuss who calls "STOP", who controls the aircraft during the stop, and the duties of each crew member during and after the stop.
If your company does not allow FOs to call "STOP" then I would suggest you keep any malfunction calls as short as possible during this critical stage of flight. Eg call "YELLOW GEN FAIL CAPTION" rather than "There seems to be a problem with one of the generators, what shall we do?"