However, I have spoken to a few folk recently about accessing their own (and others) Altmetric scores when they don't have access to a specific link or an Altmetric paper number. The easiest way to do this is via a short snippet of HTML that includes the DOI number of any paper.
Here is the HTML (DOI number is in bold):
Which generates the following badge and link:
This also allows for some interesting customisation options by modifying variables (e.g. type and/or mentions).
Altmetric actually provides a longer winded version of this tutorial on their own website, which can generate the code from a few drop down menus!