Do AWS EKS managed nodes receive security patching automatically or is it the owner's responsibility to patch?
This is likely to depend on exactly how your worker nodes are provisioned in the cluster.
If you're using unmanaged node groups then the AWS documentation here recommends that the customer upgrade their nodes manually.
If you're using managed node groups , AWS manage the node lifecyle as detailed here which I would take to mean that they will be patched and upgraded by AWS.
Of course if you're using workloads in EKS based on Fargate, the underlying hosts are managed fully by AWS, meaning you don't have to worry about patching them.