Question: Without warning, a wizard casts *telepathy* targeting a friend who is on a different continent.

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> You create a telepathic link between yourself and a willing creature with which you are familiar.
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How is it decided whether the target is willing? Does the wizard get 6 seconds to quickly introduce herself? Does the target "feel" a "friend request/phone call" and envision the caster in their head and then decide whether to let them in/answer?

I wondered if the spell assumes the target is willing until the target decides otherwise, but there is no option for the target to end the spell.

Answer: 
On further review, later in the spell description it says (emphasis added):

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> Until the spell ends, you and the target can instantaneously share words, images, sounds, and other sensory messages with one another through the link, **and the target recognizes you as the creature it is communicating with**.
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I see two reasons for that text:

1. To handle the corner case of multiple telepathy effects (you know who is who among the many voices in your head); and
2. The word "willing" is in error earlier in the spell and this sentence is indicating that the target knows who is "calling them".

Note that the sharing of thoughts is optional, so there is no risk for unwilling recipients except maybe distracting images?

I agree that RAW, you need to arrange things ahead of time (as per above), but I'm thinking that RAI you make the connection and they know who you are and can choose whether to share their thoughts.
The rules don't specify the exact mechanism, so the GM will need to adjudicate.

If the target of the spell is a PC, the GM will need to give the player enough information to decide if the player is willing.

If the target of the spell is an NPC, the GM will need to decide for the NPC.

The GM may want to establish house rules for exactly how it works, but the need for those rules is going to vary tremendously from table to table.

In many cases, the adjudication should be pretty easy.

Let's assume the target is a PC:

GM: You get a strange feeling. Your friend, Wendy the Wizard, is trying to establish a telepathic link with you. Are you willing?
Player: I...how do I know it's Wendy?
GM: You just know.
Player: How do I know?
GM: It's magic, but it's sort of like seeing someone's face or hearing their voice, you recognize them.

At this point, the player decides whether to accept the connection or not, or maybe they ask more questions.

A given table may need to have further discussion, or may not.