Does anyone know if it is required to have Pulumi Cloud or self-hosted Pulumi Cloud in order to get the Pulumi Kubernetes Operator to work?

You can use other state backends - see https://github.com/pulumi/pulumi-kubernetes-operator?tab=readme-ov-file#login-to-your-chosen-state-backend the S3 backend example (you can use others that Pulumi supports i.e. azure blob, google cloud, s3-"likes", etc)

thanks, I will take a look