The responsibility for registering trials to ClinicalTrials.gov depends upon a number of things. The Sponsor (usually the pharma/biotech/medical device company) that initiates the trial and its employees conducting the trial is the entity that registers the trial. For multicenter trials, it is often the Sponsor, but could also be someone who initiates the trial and has someone else (ie, CRO) conduct the trial. The Sponsor, however, may also designate the PI to be the Responsible Party and to be responsible for registering the trial. NIH/FDA considers holders of an IND or IDE the sponsor/responsible party (i.e., sponsor-investigator).
If your institution is only one of many taking part in a multicenter trial in which the Sponsor is a pharmaceutical company, it is most often the pharmaceutical company that will hold the IND/IDE and will register the trial. However, each academic medical center/academic research organization should verify that this is the case as stipulated in their contract with the company.
In terms of who is held responsible, FDAAA does not have a provision for this and that responsibility will most likely (or should) be determined in agreements between investigators, IRBs, academic medical institutions/academic research organizations, and sponsors, etc.