In the most common two-dimensionalist treatments, a scenario is a centered world: an ordered triple of a possible world along with an individual and a time in that world.
The primary intension of a token of 'I', evaluated at a centered world, picks out the designated individual at the "center" of that world.
So the primary intension of my use of 'I', evaluated at a world centered on Napoleon, picks out Napoleon, rather than David Chalmers.
So the primary intension of my use of 'I', evaluated at a "Twin Earth" world centered on a subject surrounded by XYZ in the oceans and lakes, picks out XYZ, rather than H2O.
We can then say that at least on the centered worlds understanding, the primary intension coincides with the "diagonal" of the two-dimensional intension (i.e. the value of S's primary intension at a centered world W coincides with the value of S's two-dimensional intension at the pair (W, W^*), where W^* is the possible-world element of W).
If scenarios are understood as centered worlds, this will be a world centered on the speaker and the time of the utterance.
On the centered-worlds version of epistemic two-dimensionalism, scenarios are identified with centered worlds.
But I will focus on the centered-worlds understanding here.
To handle these claims about self-location, we model epistemic possibilities using centered worlds.
The individual and the time marked at the "center" of a centered world serve as a "you are here" marker, which serves to settle these claims about self-location.