We had three conversations [with each other] during the week.
The phrase "with each other or one another"is a signal for a reflexive pronoun:
- We had three conversations [between ourselves] during the week. That means "We spoke to each other three times in conversations." They had a conversation [between themselves]". In other words, between themselves and not between themselves and other people.
Please note: "We had 10 apples between us." When you read that, you do not know which individual had what number of apples. The same would be true of conversations.
Question: "How many [phone] conversations did the three friends have that week?"
Answer: I don't know how many each had, but they had 5 conversations between them.
- between them refers to some number of conversations (or items) per person
- between themselves refers to conversations engaged in by two (or more) people with each other.
That said, colloquially, people do say: conversation between them, but you can see where it might be misleading with my example, and why themselves is more accurate.