Does the target of Scrying know they are being spied upon? The description says if the target knows, they can choose to fail. This implies that the target might not know they are being spied upon but it isn't clear.

I'd say no, they do not know.

Nothing says that they know, so they don't.

You can use spells like Sending before you scry on them.

Also, while not in combat someone is likely to set up a set time to scry on loved ones far away.  Works great if both can cast.

The Scrying spell doesn't say the target know about it.

Was curious because it has the target make a save. Does making a save make you aware that something is happening? Or is it one of those "You get a wierd feeling but it passes".

This just gave me an idea, where one of the bbeg’s minions gets themself captured specifically so it can then volunteer to fail it’s save vs scrying, and the BBEG can get a good look at what’s going on with the party. 

Here's a Sage Advice official ruling on the topic;

Do you always know when you’re under the effect of a spell?

You’re aware that a spell is affecting you if it has a perceptible effect or if its text says you’re aware of it (see PHB , under “Targets”). Most spells are obvious. For example, fireball burns you, cure wounds heals you, and command forces you to suddenly do something you didn’t intend. Certain spells are more subtle, yet you become aware of the spell at a time specified in the spell’s description. Charm person and detect thoughts are examples of such spells.

Some spells are so subtle that you might not know you were ever under their effects. A prime example of that sort of spell is suggestion. Assuming you failed to notice the spellcaster casting the spell, you might simply remember the caster saying, “The treasure you’re looking for isn’t here. Go look for it in the room at the top of the next tower.” You failed your saving throw, and off you went to the other tower, thinking it was your idea to go there. You and your companions might deduce that you were beguiled if evidence of the spell is found. It’s ultimately up to the DM whether you discover the presence of inconspicuous spells. Discovery usually comes through the use of skills like Arcana, Investigation, Insight, and Perception or through spells like detect magic.