I can't find an official report, but from interviews I get the impression that multiple people within the Celtics organization have a vote in roster composition, including Brad Stevens!
If this is true, than I find that really disturbing. You can't let a coach be involved in big management decisions. That's asking for trouble. I think letting a coach have a vote in the drafting process (for instance) is an absolute no-go. Even as an advisor I'd rather trust my scouts. You need authority, responsibility and accountability within an organization. You can't mix them up or you'll create a mess.
When you're not able to sustain those values at the top - macro level (management), then it's not a surprise that you get the same problems at meso level (coaching) and micro level (players).
Let me tell the story of another legendary sports club/franchise:
A couple of years ago (November 2010), there was a 'Velvet Revolution' at Ajax (Football Club). At every position in the organization it was forced that people would need to be replaced by ex-footballers, who would have the knowledge to get the club back on track instead of 'institutionalized' people.
Great idea and only the legend Johan Cruijff could have enforced it, without even being a part of the actual organization. However, he had so much trust in the capabilities of educated ex-footballers with loads of experience as footballers (not as managers) that he created the idea of a 'Technical Heart'.
This 'Technical Heart' consisted of a number of people within the club (general manager, technical director, head of scouting, head of youth academy, development specialist, assistant-coach, head coach). This 'Heart' of the club came together (at least) once a week to manage all the new development programs, playstyle philosophy, trainer evaluations, player evaluations, scouting, transfer policy, you name it.
This 'Heart' had all the power and everybody had an equal vote, this was even written in their contracts. Since the members all had great reputations, most had known each other before as players (teammates), it was expected that difference of opinions could easily be solved. Despite all the sincere intentions, things quickly derailed. It turned out that some members were more radical (wanting to fully focus on youth development) than others (listen to the head coach demands for expensive players).
Soon the head of the youth academy departed, even though he was one of the key figures in enabling the 'Velvet Revolution'. After a while the assistant-coach (who didn't want the responsibility of head coach or technical director), became unhappy because he thought he didn't have enough influence on the training field. So he basically used the 'Technical Heart' to increase his influence, what followed was a big clash between him and the head coach. He demanded the head coach to be sacked and that was what happened.
A year later he clashed with the technical director and he almost succeeded to get him out, but at the last moment the supervisory board stepped in and asked the assistant-coach to leave and he did. After that authorizations, boundaries and clear lines within the club were established. Managers and directors got back decision power. And passersbys like coaches/trainers/specialists/scouts would never have direct influence on policy again.
Long story, but the lesson was learned: always have clear defined roles on authority, responsibility and accountability within the organization.