I would have a custom made combo trellis / lattice structure built on two sides ( from the stone wall around to the step). I would have something substantial, not the flimsy lattice you see everywhere.

I can't really get a good sense of what I'm looking at here. This seating area seems like it's way too far away from your house (?) I would rethink the entire back yard plan before I spent money building any type of structure so far away from what appears to be your house.

Privet is pretty fast growing. If you're in planting zone 7B or further south, you could use Italian Cypress as well. This link may be helpful. https://www.enkiverywell.com/fast-growing-shrubs.html

Inspiration:

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This is a beautiful patio! You must be so excited! I would consider an Arborvitae hedge, maybe "emerald green variety". A good local nursery can advise on which is the fastest growing. Arborvitae is affordable and would likely grow faster than boxwood, especially if you fertilize yearly. I would place the hedge staggered about 5' from the patio on the street side. Because of the slope, you will want to get as tall a scrub as possible but it is easy to find a 6' plant for a good price. You might want to buy about 20 of them and stagger them in 2 lines on the street side. The hedge would be 15' high before you know it and would provide total privacy. If you like, you could add a bed of hosta between the arborvitae and the patio. It also grows very quickly. People often give them away and they can be easily divided. Enjoy your lovely patio!

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