Firestore now has two functions that allow you to update an array without re-writing the entire thing.

Link: https://firebase.google.com/docs/firestore/manage-data/add-data, specifically https://firebase.google.com/docs/firestore/manage-data/add-data#update_elements_in_an_array

Update elements in an array

If your document contains an array field, you can use arrayUnion() and arrayRemove() to add and remove elements. arrayUnion() adds elements to an array but only elements not already present. arrayRemove() removes all instances of each given element.

Edit 08/13/2018: There is now support for native array operations in Cloud Firestore. See Doug's answer below.

There is currently no way to update a single array element (or add/remove a single element) in Cloud Firestore.

This code here:

firebase.firestore()
.collection('proprietary')
.doc(docID)
.set(
  { sharedWith: [{ who: "third@test.com", when: new Date() }] },
  { merge: true }
)

This says to set the document at proprietary/docID such that sharedWith = [{ who: "third@test.com", when: new Date() } but to not affect any existing document properties. It's very similar to the update() call you provided however the set() call with create the document if it does not exist while the update() call will fail.

So you have two options to achieve what you want.

Option 1 - Set the whole array

Call set() with the entire contents of the array, which will require reading the current data from the DB first. If you're concerned about concurrent updates you can do all of this in a transaction.

Option 2 - Use a subcollection

You could make sharedWith a subcollection of the main document. Then adding a single item would look like this:

firebase.firestore()
  .collection('proprietary')
  .doc(docID)
  .collection('sharedWith')
  .add({ who: "third@test.com", when: new Date() })

Of course this comes with new limitations. You would not be able to query documents based on who they are shared with, nor would you be able to get the doc and all of the sharedWith data in a single operation.

Here is the latest example from the Firestore documentation:

firebase.firestore.FieldValue.ArrayUnion

var washingtonRef = db.collection("cities").doc("DC");

// Atomically add a new region to the "regions" array field.
washingtonRef.update({
    regions: firebase.firestore.FieldValue.arrayUnion("greater_virginia")
});

// Atomically remove a region from the "regions" array field.
washingtonRef.update({
    regions: firebase.firestore.FieldValue.arrayRemove("east_coast")
});