Keep using private by default. If a member shouldn't be exposed beyond that type, it shouldn't be exposed beyond that type, even to within the same project. This keeps things safer and tidier - when you're using the object, it's clearer which methods you're meant to be able to use.

Having said that, I think it's reasonable to make naturally-private methods internal for test purposes sometimes. I prefer that to using reflection, which is refactoring-unfriendly.

One thing to consider might be a "ForTest" suffix:

internal void DoThisForTest(string name)
{
    DoThis(name);
}

private void DoThis(string name)
{
    // Real implementation
}

Then when you're using the class within the same project, it's obvious (now and in the future) that you shouldn't really be using this method - it's only there for test purposes. This is a bit hacky, and not something I do myself, but it's at least worth consideration.