How could England and Scotland be independant under one crown?

When the crown of England and Scotland formed in 1603 by king James how did the two countries remain as independant, sovereign states? It's not like now when you could union with the monarchy but remain politically independant.
1603 was a time when the monarchy had most of the political power and influence, the subjects and parliment very little. Surely in 1603 uniting the crown's would mean uniting the parliment because the crown would be the head of parliment of both countries. Can anyone shed some light on this.

What is a VPN?

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