When talking to the townspeople, do I have to finish the conversation to earn full friendship points?

Solution 1:

It looks like simply talking to them will give all 20 Friendship points, even if you don't finish the conversation.

See this thread:

  1. If a villager has a conversation that lasts multiple clicks, do you need to go through the whole thing, or just the first one?

Just the first thing. As long as you interact with them to get the dialogue box up, you get the points.

In this wiki chat log, it looks like someone had been wondering the same thing as you:

So it's apparently either +10 or +20. Any info on why? Any speculations? Any chance it could have something to do with whether or not you "finish" their dialogue for the day (for example, talking to them once when they have multiple dialogues gives +10, but talking to them and finishing the dialogues gives +20)?

It appears that this has been figured out as the wiki now explains the reasoning behind differing Friendship point gains:

talking to villagers (+10 if the villager is in the middle of a special animation, like exercising or using a camera, otherwise +20; can only score once per day);

Thus, as long as you converse with a villager at least once a day and they're not in the middle of something special, you'll get +20 Friendship points, even if the conversation could have kept going.