There are many test networks on Tezos.
What are the active and soon to be active public ones? How do we participate? Are there accessible public nodes?
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As commented above, https://tezos.gitlab.io/introduction/test_networks.html lists the current networks: mainnet, Edo2Net (mainnet test), Florencenet (next protocol, if it passes)
Every network is public and both test networks have free tokens from the faucet. You can participate by downloading/running the latest tezos software and using the
--network flag to specify the network to which you want to connect.
You should definitely have a look at the Teztnets. This framework has been developed to help everyone easily join testnets for Tezos. Having Docker pre-installed is enough and the simple commands of the website (depending on which testnet you which to take part in) will automatically let you join the effort.
Although, the link in the documentation is correct, this website actively keeps track of the current testnets and serves to automatically spawn daily and weekly testnets to challenge the latest codebase.