For Cloud JIRA, you will be able to see the logs for actions performed by our add-on by looking at the JIRA audit trail. On the current version we only do "reads" which are not logged by default.
If you want to see a detailed log of calls made to your JIRA instance you should contact Atlassian support. See AC-1858 from the Atlassian Connect Issue tracker.
JIRA server plugin actions are not logged by default but an info log level can be enabled for the add-on. This will log a lot of information into your JIRA instance log file. Please note that this logging level is lost when you restart the instance.