Amalfi, Sicily, or Sardinia in August for 1-week?

A lot really depends on what you mean by beach--chairs + umbrellas? sand v pebble? etc. But I would omit the Amalfi Coast if I were planning an August trip. Book your accommodations asap.
It is the highest of high season for anywhere coastal, but at least on a large island you can spread out more.

For beaches, I like the Cilento coast south of Salerno, Puglia, the Marche coast near Ancona, and southern Tuscany.

Yes, the Amalfi Coast will be absolutely jammed in August. It's hard to navigate any time of year, but absolutely packed in summer. We were just there in March and even the private drivers we used were telling us that summer is just a mess. If you're looking for something quieter, Sicily or Sardinia seem a better option (I would have to believe those would be cheaper too).

I don't know about Sardinia, but Sicily will probably be miserably hot in August. And one week is a really, really short time to spend there. It's a large island with top-flight sights scattered all over. I usually recommend at least two weeks in Sicily, and that's assuming you'll be driving part of the time.

I've never been to the Amalfi Coast in August, but it's packed all other summer months, so August will be worse. I'm sure you can find restaurants, although there might be waits. The Sorrento peninsula is very rocky so more than beaches you will see cliffs with some small pebbly coves.

Sicily and Sardinia will be also full of tourists, especially beach localities, but some places will be worse than others. Prices will be highest of the year at both locations.

The best beaches of the Med are certainly in Sardinia. But you'll need a car for sure and renting a car in Sardinia in August after Covid cost an average of $700 to $1000 a week. The North half of Sardinia is best, in my opinion.

Sicily offers more in terms of history and cultural sights. They have some great beaches too. With only a week you can cover very little of the wealth of things to see in Sicily, so you will have to concentrate on a smaller area. I think the better beaches are on Western Sicily, e.g. Zingaro Marine Reserve, San Vito lo Capo, just to name a couple. Even Palermo has a great beach in the district of Mondello, just north of the city center. The smaller islands around Sicily (Aeolian, Egadi, Pelagie islands) have great beaches too, but the logistics to get there takes a bit of time.

They are all great places to vacation, but if you go in August, it will be full and pricey. That's when most Italians (and Europeans) head to the beaches, so no matter where you go, it will have people.

Have you thought about flying (or taking the train) to Naples and then ferrying to Ischia? It would be a chill place to spend the week, at least compared to Amalfi/Cinque Terre. Beautiful beaches, natural mineral baths, terrific food.

We landed on Sardinia 2 nights ago, in Cagliari. We’re going back to that city in a week and a half, after driving to other parts of Sardinia. We’re currently spending our second of three nights in Sant’ Antioco, a small city on a small island with the same name, connected to the main island of Sardinia by a causeway. They just had a big annual festival today, but we’ve heard only two other English speakers, and they clearly weren’t native speakers, from their accents. There was also a car parked along a street tonight with UK plates. Maybe they have a second home here, or perhaps they just drove a long way from home. I get the impression there are a few Brits who come to Sardinia, and apparently few people who speak German, but relatively few tourists. Maybe the tourists on Sardinia are other Italians.

Very few residents here appear to speak or understand much English. That includes people who run restaurants, shops, market stands, and car rental desks. If you want some place that feels off the beaten tourist track, especially for Americans, Sardinia would seem to be a good candidate. Well, in April for sure, but in August, I can’t personally guarantee.

We were in Sicily one December, almost empty, and by the Amalfi Coast on the same trip, with lots more people. Again, I can’t say what you’d find in August, but I would venture Sicily would be less crazy packed than mainland Italy. Sardinia likely even less busy, however.

You might also consider Greece, either on the Mani Peninsula at the southwestern end of the Peloponnese, or on Crete. When we vacationed on Crete in September (I know, not August), there were lots of people at the beach town of Agios Nikolaos, but almost nobody else at the black sand beaches of Kato Zakros. There were some people, mostly Germans, on vacation at Pitsidia (just inland from the beach), but not a huge, or even large amount.