I would say that grammmatically speaking this is a complete sentence. As you noted, ‘tell’ can be intransitive therefore it doesn’t need an object.

By itself the sentence perhaps doesn’t represent a ‘complete thought’. However, sentences do only have meaning in the context of the passage of which they form part.

Surely the text below has meaning – and represents a complete thought.

“When they first met, neither knew the other’s name. They told each other.”

When the verb has no object for the word, there is no noun to receive the action of the word, then the verb is intransitive.

example: I coughed.

When a verb has an object that receives the action of the verb, we say that the verb is transitive, therefore you need something (an object) after the verb in order to complete the thought.

example: I wrote a poem.

“I went” is transitive because you require an object to complete the thought.

N.B. “I went to the store yesterday”: went is still TRANSITIVE because “to the store” is a prepositional phrase and “yesterday” is an adverb.

Similarly, “They told each other.” is not a complete thought because “told” is a transitive verb. You need an object to complete the thought. The told each other something.

I have posted a handy link below which explains well and has exercises also.

Cudos to Becca Lynn. Joseph Smith said that he could have provided a better translation or interpretation to the KJV but that it was good enough for his purposes. Smith started on an inspired translation but never completed it. We have pieces of it that were preserved by the Reorganized LDS started by his son, Joseph III. Those exerpts were incorporated into the current LDS publication of the KJV of the Bible. One of the things I found ironic was when I was reading the Portuguese translation of the Bible. I took three volumes and compared the same verses in each one and found them worded differently. The new English translations of the Bible do the same thing. The different wordings can and do yield different interpretations or at least nuances to the verses. That is another reason why the Bible states that the scriptures are of no private interpretation. Meaning that we should always seek the inspiration of the Holy Ghost when reading the scriptures so that we can have the Lord’s understanding and interpretation thereof. 2 Pet. 1: 20 20 Knowing this first, that no prophecy of the scripture is of any private interpretation.

I see both points here, although using the intransitive verb you are correct indirectly. But in proper english standards this is a phrase or fragment.

Although “to tell” can be intransitive, the sense implies it is being used as a transitive verb in this instance. They do need to tell each other something.