SQAD, you are not correct when you stated,
"But if both are vaccinated, then they can't get the disease and hence cannot spread it to an unvacinated society."
From the Government's website
"Will the vaccine protect you?
The COVID-19 vaccine that you have had has been shown to reduce the chance of you suffering from COVID-19 disease. Each vaccine has been tested in more than 20,000 people in several different countries and shown to be safe.
It may take a week or two for your body to build up some protection from the first dose of vaccine. Like all medicines, no vaccine is completely effective, so you should continue to take recommended precautions to avoid infection. Some people may still get COVID-19 despite having a vaccination, but this should be less severe.
Can you give COVID-19 to anyone if you have had the vaccine?
The vaccine cannot give you COVID-19 infection, and a full course will reduce your chance of becoming seriously ill. We do not yet know whether it will stop you from catching and passing on the virus, but we do expect it to reduce this risk. So, it is still important to follow the guidance in your local area to protect those around you.
To protect yourself and your family, friends and colleagues you still need to:
practice social distancing
wear a face mask
wash your hands carefully and frequently
follow the current guidance"