use a model without publishing?

Is it possible to use a model you trained without publishing it?

If you use a local installation of Stable Diffusion, it is possible. If you mean in Civitai generator, I don't think it can be done as such model wouldn't go through necessary security checks (code and contents).

If you want to run SD locally look up Stable Diffusion Webui (Automatic1111). There are tons of guides on how to set it up. That way you can actually test your model. Civit only allows you to use models that are public and have the commercial use for the CivitAI generation service enabled.