Teresha, welcome to WmW.
Macguru is correct, in my experience you do not want the to be orphaned or left to stand alone, they do not get their strength from that. So, to take your question a bit deeper, first make sure you have the internal linking correct. If set up correctly all the pages within the sub-domain or canonical can be linked together. If not and they are not on theme then something isn’t right and you want to look at that. You will also want it legitimately linked to and from the homepage and then from any other canonicals within the structure of the site that relate.
Once you have this internal linking built in strongly so the site is supported you can begin to look for linking partnerships, make directory submissions and find themed portals or vortals that support your separate canonical themes. In essence then your whole network grows, not the specific canonical.
>…would see a link to www.subdomain.mysite.com as a link to www.mysite.com and vice versa…>
No, in most cases they are looked upon as separate sites but, if the internal linking is strong and correctly implemented than the canonicals will benefit from the separate incoming links they each receive.
>… if the sub domain was given 20 links from sites relevant to its topic and the main site on the root domain is granted 20 links from other sites relevant to its topic would it count as 40 links in total or are they counted separately…>
In simple terms, no. If implemented as Macguru and I have suggested than as a whole, yes.
If you are developing a site with canonicals then make them work for you. If you do not tie them in correctly you can actually lose benefit from them. You would be dealing with several sites and not one strong network. Good luck, I hope this helps.