Wood (uncoated) trellises.

Climbing hydrangea sends branches out about four feet. That gives each trunk an 8’ wide drape. Your cable (and floral’s swag) idea sounds unappealing to me.

Climbing hydrangea? I didn’t see any reference to them in the OP so I’m not sure of the relevance. They certainly wouldn’t work on swags.

No but climbing hydrangea is a vine, isnt it? I just don't like the idea of swags. It just seems like a laundry line in this case. Maybe one could pull it off in an urban English setting but I can’t see it in a wooded suburban setting unless your objective were to goad the neighbor to erect his owm fence. Id be interested in seeing a picture of a succesful scheme

OP refers to ’a couple of oaks’, so I wasn’t envisaging woodland, just two or three trees at the edge of the yard. Free standing swags would be a way of achieving what the OP was asking about, given sufficient light. I wouldn’t necessarily endorse it, although they are stunning if well maintained. I don’t associate them necessarily with urban gardens. Much more a cottage/country vibe. Since Climbing Hydrangea is self clinging in the same way as Ivy it would be unsuitable for the same reason I gave in my initial response.

Trellises.

Except for some things that just sound not very attractive, a lot depends on the actual site- the oaks, likely root zones, wher the property line is, where the shade is. And if you’re looking for 10” of height of major screen- that’s hard. Or whether a kind of soft blurring at walking height level helps.

I would never plant ivy ( I inherited it at my house & gradually removing) but especially not to train as climber. Even if you could establish an attractive trellis of obelisk, over time by allowing it to climb & develop its trunk it will morph into flowering/ fruiting form and be more likely to spread willy- nilly. In addition the part you planted will send our runners constantly.

Depending on zone & light and the oak roots, you might be able to establish some shrubs or small understory tree(s) further into yard that add sense of depth and seclusion when your in the yard of sitting.
Then a common question is, what are you screening- if patio for instance, then creating screwing right there can be effective.

Whoops.
“ creating screening” !
Gotta say, auto- correct is … inventive.

Hydrangea wouldd cling to tbe oak trunks and brsnch outward from there.

FYI, it's woodland. The oak trees are the biggest trees, but there are maples and firs too.

The big issue really comes at the end of the fall and during the winter. It's not much of an issue otherwise because the deciduous trees have enough leaves to hide my neighbor's under-deck.

I ended up removing a few branches from the deciduous trees, dumped between 2 and 3 tons of top soil retained by small boulders and planted a 4 foot White Pine + a 5 foot Norway Spruce. Hoping they get big enough within 10 years...

I even thought about dumping a 20-30 tons of top soil to create a mound + big-ish boulders to retain all of it, but my wife thought we could do something else with that kind of money and that I'm crazy.

If you dumped that soil on tree roots, the trees will die. It'll take a few years, but they'll die.