What is the meaning of `private` attribute in `tfstate` (for a DynamoDB instance)?

The "private" property is a place where providers can retain any metadata they need for internal lifecycle tracking, separately from the actual data in "attributes".

From Terraform Core's perspective it's just an arbitrary sequence of bytes, base64 encoded. You can base64-decode it to see what the provider is storing there.

A provider is free to store whatever it wants in principle, but in practice today this is most commonly used by the Terraform SDK to track schema versions. That is likely what you'll find if you decode this one, in JSON format. The Terraform 0.12 state snapshot format has a first-class property for schema version which you can see further up in that object, so in this case that data is redundant but the SDK is retaining it in two places so that providers can remain compatible with Terraform 0.10 and 0.11.

"private" in this context means "for use by the provider only", not "secret". Therefore from the perspective of the detect-secrets tool this is an unfortunate false-positive. It is possible in principle for a provider to store private data in there -- just as it's able to do so in the "attributes" object too -- but that's not what this property is designed for.