Share a cookie between two websites

Question

I have built a website (A) which logs in to and retrieves customer data from a separate web service.

The organisation that owns (A) also has a website (B) which has

Answer 1

Cookies are only accessible to a single domain that they are set to.

I believe if you are using two sub-domains on the same domain it would be possible to share the cookies, however the browser doesn't send cookies set on one domain to any others.

Edit: You also want to avoid storing large amounts of data in a cookie. Is there perhaps the chance you could create an api that site B could query with javascript?