How to copy files between Google Cloud Storage with minimal cost?

I have been granted access to a Google Cloud Storage and I want to copy some files (maybe large) from it to my own Google Cloud Storage bucket. I’m looking for a method that has the lowest cost.

I know I can use the gsutil cp command to copy files, like so:

# on my own server
gsutil cp gs://source_bucket_name/file gs://target_bucket_name/

However, I’m not sure if this will double the traffic charge (e.g. whether it will incur traffic charges for downloading from source_bucket and uploading to target_bucket, or gsutil will process without download it). I’m wondering if there’s a better way to copy the files I need while minimizing the cost.

Lastly, I am not sure what kind of additional costs would it incur?