As for why the transaction logs is not truncated, check the column log_reuse_wait_desc in sys.databases. There could be other reasons why the log is not truncated.
Apart from that, I can't say that I am happy with this arrangement. In case there is a disaster, and a database needs to be restore to a point in time. Do you know how to do that with help of the full backup from the third-party tool and the logs from the log-shipping?
If you are confident with that process, fine. If you are not - turn off log-shipping and enable log backups in the external tool immediately!
Log backups will truncate the log, provided that nothing else blocks truncation. There is one exception: You take the log backup with COPY_ONLY. Which is good so far that it does not interfere with with the third-party tool. But if the third-party tool truncates the log, the log-shipping will soon have a broken log chain.
After reading https://dba.stackexchange.com/questions/224711/does-mssql-log-shipping-truncate-the-transaction-log , i realize that truncating the transaction logs seems to not be part of the log shipping process by design.
That post talks about the size of the log file not shrinking. Truncation only means that space in the log is freed up for re-use, but the file size remain the same, unless you actively shrink it.