One VPS, multiple magento installations. Do I need to install composer in each domain?

I am trying to update magento from 2.3.4. to 2.4.5.
In the past I had installed composer as root user on my VPS.
This is not advised and can cause issues with reading/writing of files.

• Can/should I install composer in each magento installation?
• Is it possible to just remove the old global composer installation and replace it by a magento domain based installation? Will magento be bothered by that?