Are there free European E-commerce platforms?

Yes. PrestaShop, Saleor and Shopware are free or have a free plan, according to the pricing each company publishes. Free plans usually carry limits; each product page links to the company's terms.

Which European E-commerce platforms are open source?

PrestaShop, Saleor and Shopware publish their source code under an open-source licence. Each product page shows the licence and where the code is.

Which European E-commerce platforms can you host yourself?

PrestaShop, Saleor and Shopware can be run on your own servers, according to their own documentation.

European E-commerce Platform Alternatives | European Online Store Builders

E-commerce Platforms alternatives at a glance

European E-commerce Platforms alternatives compared by country, pricing, free plan, open source and self-hosting
Product	Country	Pricing	Free plan	Open source	Self-host	Replaces
PrestaShop	🇫🇷France	Freemium	Yes	Yes	Yes	Shopify, BigCommerce +3
Saleor	🇵🇱Poland	Freemium	Yes	Yes	Yes	Shopify, BigCommerce +3
Shopware	🇩🇪Germany	Freemium	Yes	Yes	Yes	BigCommerce, Wix +3