Third-party testing means the analysis was performed by a laboratory independent of both the manufacturer and the seller — a party with no stake in the outcome.
The value is structural rather than technical. The instruments and methods are the same ones an in-house laboratory would use; what changes is who is accountable for the result. The same HPLC run means more when the organisation reporting it is not the organisation being evaluated by it.
It is only checkable if the resulting certificate of analysis is actually published, names the testing laboratory, and identifies the batch it belongs to. A claim of third-party testing with no accompanying document is an assertion, not evidence — which is why Echo Peptides publishes the certificates it holds and states plainly which compounds do not yet have one.