No caption needed or no needed caption

The phrase "No caption needed" is not a complete grammatical sentence. If you have ever looked out at the wings of an aeroplane in which you are travelling you will probably have seen the words "No step". If you have ever had the misfortune to be questioned by the police in the UK (no Fifth Amendment in that country) you might have been advised to reply "No comment" to some questions. If you ride around the UK on a pedal cycle you will frequently find notices proclaiming "Cyclists Dismount".

In all such cases the meaning is clear but no grammatical sentence has been used. Nevertheless a complete thought has been expressed.

But once you say "no needed caption" you have created a noun phrase. Immediately your audience is asking " and so...?". You have not stated a complete thought.

"needed" is a past participle. Like prepositional phrases, past participles act much like adjectives, but often come after the noun or noun phrase they modify. Here, the past participle is used for the passive voice, with the copula omitted. In passive voice, the past participle follows what would be the object if it were active voice.