What does the "private" modifier do?

It's for you (and future maintainers), not the compiler.

There's a certain amount of misinformation here:

"The default access modifier is not private but internal"

Well, that depends on what you're talking about. For members of a type, it's private. For top-level types themselves, it's internal.

"Private is only the default for methods on a type"

No, it's the default for all members of a type - properties, events, fields, operators, constructors, methods, nested types and anything else I've forgotten.

"Actually, if the class or struct is not declared with an access modifier it defaults to internal"

Only for top-level types. For nested types, it's private.

Other than for restricting property access for one part but not the other, the default is basically always "as restrictive as can be."

Personally, I dither on the issue of whether to be explicit. The "pro" for using the default is that it highlights anywhere that you're making something more visible than the most restrictive level. The "pro" for explicitly specifying it is that it's more obvious to those who don't know the above rule, and it shows that you've thought about it a bit.

Eric Lippert goes with the explicit form, and I'm starting to lean that way too.

See http://csharpindepth.com/viewnote.aspx?noteid=54 for a little bit more on this.