In a self-hosted setup, the organization runs the model itself, on servers it owns or rents, and not through a vendor's API. It usually uses an open-weight model, and the team is responsible for the whole stack, including the GPUs, the serving software, scaling, updates, monitoring and security.
The reasons to do it are control and privacy, since data does not leave the organization's environment, along with freedom to customize and fine-tune the model, avoid dependence on one vendor's pricing and rules, and in some cases lower costs at high and steady volumes.
The drawbacks are real. Specialist skills and hardware are needed, usage may be unevenly spread so that expensive chips sit idle, and the best closed models are not available for self-hosting. A hosted API charges only for what you use, while your own servers cost money whether or not they are busy.
Estimate the total cost, including staff and idle capacity, against API pricing at your expected volume, and be sure about your security and compliance needs.