Their cheapest tier is free, they lose money on that of course. And spend a lot of money training new models.

Anthropic has said they have made money on every model so far, just not enough to train the next model, which so far has been much more costly to train every generation. At some point they will probably train an unprofitable model if training costs keep rising dramatically.

OpenAI burns more money on their free tier and might be spending more money building out for future training (I don't know if they do or not) but they both make money on their $20 subscriptions for sure. Inference is very cheap.