Facilitating the responsible access to genomic research data is an emerging ethical and scientific imperative. Data Access Committees (DACs) assess the ethical footing and scientific feasibility of the data access requests and evaluate the qualification of applicants to ensure they are bona fide researchers. Through semi-structured interviews, we explored the opinions and experiences of 20 DAC members and experts concerning the users' qualification criteria and mechanisms to hold users accountable. According to our respondents, such evaluation is necessary to ensure applicants are trustworthy, meet a certain level of expertise or experience and are aware of the rules and the associated concerns with genomic data sharing. The respondents noted, however, that the qualification criteria are fragmented or are poorly delineated at times. Thus, developing qualification criteria seems vital for an objective, fair and responsible access procedure. Similarly, the access review will benefit from using common ways of verifying the users' affiliations. Furthermore, some DAC members expressed concern over the uncertain oversight of downstream data use, in particular where data are shared across borders. DAC members and experts did not consider current sanctions and enforcement procedures to be crystal clear. Therefore, data sharing policies should address this gap by establishing proportionate sanctions both against data producers and data users' non-compliance. Users' home institutes will need to have an active role in keeping oversight on the downstream data uses, considering their ultimate responsibility if wrongdoings happen.