I, personally, would never store my connection strings in my machine.config file, only ever in the web.config.
You say it's a dedicated server, but is this server dedicated to a single web application?
If not, you've now got a single file (machine.config) that's effectively sharing configuration data for multiple (probably un-related) web applications. Depending upon the number of those applications, and if you ever needed to move them to another server, using the machine.config could get very messy.
Even if the server is dedicated to a single web application, you'll probably be performing your development and testing on other machines. Since the machine.config is not normally a file that's included within the file-set of a ASP.NET web application project, you will probably have to go out of your way to deploy the machine.config to each of the various dev/test/production machines, and ensure that other (non-connection string) related configuration within them is correct for that machine.
Using the web.config to store database connection strings makes perfectly logical sense and is entirely expected by almost all ASP.NET developers, even if you have multiple applications that will use the exact same database on the exact same database server. Keeping this configuration in the web.config of each application allows each of those applications to be more self-contained and not reliant upon some other "outside" file.
I generally view the machine.config and something that the framework itself uses, and it belongs on a machine and is specific to that machine. I very rarely go in and touch it myself. I view the web.config as a file that's part and parcel of your web application project and "moves around" with that project as it moves and is deployed to different machines.
Also, don't forget that a lot of what's defined in the machine.config can be "overridden" on a per-application basis by redefining certain configuration elements within a specific application's web.config file.
Of course, there are a few valid reasons to editing/changing your machine.config file (such as multiple web servers in a web farm will probably need things like encryption/decryption keys within the machine.config to be synchronised), however, I don't believe database connection strings are one of those valid reasons.