A patient can be sedated and paralyzed on any mode, or be in such a poor state that they are neither adding rate nor triggering a breath. On a conventional vent, if you are using APRV/Bi-Level, you DO want them to breathe over the vent.
On the oscillator, you never want them to breathe over the vent. Same for VDR and I think the Bronchotron.
For babies on jets, the rate is usually 360 or 420, and they can also have conventional breaths (sometimes called sigh breaths), as well as breathe over the vent but they shouldn't be fighting it. If they are asynchronous, that is my cue to speak to my RN and MD.
On a conventional ventilator in AC/PC, AC/VC, PRVC, SIMV, and any other mode except for PSV/Spont/PS/CPAP (which don't have rates, only apnea settings), the patient can trigger or not and add rate or not. If the rate is set on 12 and they are breathing 12, they aren't adding rate but they could be triggering.
Every ventilator uses different names for modes of ventilation. If you tell me the vent that you have and any specific questions, I'd be happy to help.
At my hospitals, if the patient is on the jet, oscillator, or VDR, the respiratory rate is left blank on vital signs and under oxygen device, you would select the appropriate piece of equipment. Only the RTs, under vent assessment would put the Hz and in the case of the jet, if any sigh breaths are added.