特に認証をしない匿名ユーザ向けの手順です

TL; DR

https://qiita.com/k_hoso/items/7f2763421dd37a5e2cd4

手順

IdentityPoolId から IdentityId を取得

curl -X POST \
    https://cognito-identity.ap-northeast-1.amazonaws.com \
    -H 'Content-Type: application/x-amz-json-1.1' \
    -H 'x-amz-target: AWSCognitoIdentityService.GetId' \
    -d '{ "IdentityPoolId": "ap-northeast-1:12341234-4321-4321-4321-123412341234" }'

↑で取得した IdentityId から Credentials(AccessKeyId, SecretKey, SessionToken, Expiration) を取得

curl -X POST \
    https://cognito-identity.ap-northeast-1.amazonaws.com \
    -H 'Content-Type: application/x-amz-json-1.1' \
    -H 'x-amz-target: AWSCognitoIdentityService.GetCredentialsForIdentity' \
    -d '{ "IdentityId": "ap-northeast-1:abcdabcd-4567-4567-4567-abcdabcdabcd" }'

以上。