He doesn't do any salary cap stuff and he doesn't handle any contracts (except maybe to wine and dine somebody in an effort to get them to sign for less.)

The scouting department will compile reports on all NFL players (even if they aren't free agents), all college players (even if they aren't likely to be drafted) and even on every player within the organization, and they will give those reports to the GM and his staff (which sounds like its going to be the Goodmans).

The following is speculation, but probably not far from how they do it:

The gm and his staff will go over all the reports and seperate them into "top" prospects and "everyone else."  They will individually evaluate every top prospect with Shanahan.  The everybody elses will get sifted further into groups like "perfect fit for Denver", "area of immediate need for Denver", "bad fit for Denver", "Not a position of need for Denver,"  etc.  Those groups will get categorized even further with headings like "injury issues,"  "contract issues", "should come cheap", "character issues", etc.

Shanahan will want to look at every prospect from some group (such as the "no issues/perfect fit for Denver" group).  From the rest of the groups, it is up the the GM and his staff to sift out the value and put the dossiers on Shanny's desk.

The actual process of "looking at a prospect" will involve something along the lines of:  review recent tape of the prospect, breaking down their play at their position and other situational factors based on Shanahan's interest and needs for the player.  The front office staff also consists of many scouts and media employees who are responsible for watching these players in games and taking notes, as well as interviewing the players and people who knew the players (for example, when Marcus Thomas was drafted, who had drug issues, the Broncos sent someone to his hometown in Florida to interview his friends and highschool and college coaches).  These notes and personal information go into the players files for Shanahan's review.

Once the Shanahan has looked at all the files on his desk he will prioritize what he wants, and the cap gurus and the GM will start setting up meetings with agents, visits to the team facilities, and start preparing "what if" scenarios for the creation of the new contracts.  At this point is where you start to see players cut and contracts reworked that are currently with the team as the Broncos prepare to absorb the new contracts.  While the GM and his staff handle the majority of work with the new signings, one vital aspect is when the player gets to meet with Shanahan and discuss his role with the Broncos.  This meeting and the finalizing of the money value of the contract are far and away the two most significant aspects of the process.

As players on Shanahan's "want" list are acquired or scratched off, they will constantly reevaluate the list, removing players that aren't feasible and adding players that become valuable with the signing or absence of someone else.  Shanahan is probably kept abreast of this process, but with all his other chores it is doubtful that he actively participates much at this point, except to constantly be giving "Final Say."

One aspect that has been left out above are player for player trades.  When the Broncos finish a season they conduct waht are called "exit interviews".  Basically, no one leaves Dove Valley until they have completed their exit interview, which is a meeting with Shanahan and his staff (including the rest of the coaches) where they look at tape of the players season with the Broncos, address issues that they want the player to work on in the offseason and give a heads-up to the player about his future status with the team (like, expect to be back here, or You're agent will be busy this offseason).

So you can see that Final Say is probably less about chiming in and having a say about every aspect of the rganization as it is about directing the policy and direction of an organization.  In reality, there is only so much Shanny can do, and only so many Final Says that he can hand out in a day.  He is relying on the members of his staff to present the proper choices between which he must choose and give his Final Say.  If they present him with A and B but neglect to show him C, then his power to make the best choice is undercut.

I would also note that the Broncos probably employ an independent agency that acts as a sort of auditor of different aspects of the organization.  Such as, every year, they probably have someone (not associated with the organization, and not in a position to be influenced by the organization) look over all their previous years scouting reports and diagnosises and compare them to what actually panned out for all those prospects.  The auditor will red flag inconsistencies and make reccomendations to the Broncos, along the lines of "This scout routinely mis-scouts QBs," or "This scout doesn't do a very good job of accurately assessing the teams on his beat," or "you need a stronger scouting presence in the college WAC conference, a lot of talent is slipping through the cracks there."

A longer answer than you wanted, I'm sure. :)