Yup, all nodes on the network should see transactions that are sent, blocks that are generated, etc. Right now the only functionality normally available to the regtest network that isn't available is backing up and encrypting wallets, and manually adding nodes (BTCFiddle has its own commands for managing nodes).
Over the weekend I threw together a site that allows you to easily use a regtest network to try out transactions and learn about how the Bitcoin protocol works. It's probably not completely stable right now, but I think it's a useful tool to have, and it would be great to get some quick feedback and see what other people think. Thanks for taking a look!