Depends on the ad. Some I've seen give different promocodes on different media/ads, then count codes used. Some have a weaker correlation in sales made vs ads made, where you match jumps in sales to as start dates to get rough estimates. Some just do surveys, like you said.

If its a link very often there is a part behind the normal link to a site that tells the website where you came from. It is usually something like reddit(dot)com/r/eli5/thispost?asdhjahjsdjkasdjghk the part behind the question mark then could tell me you got to this post through an add on facebook or google or what ever (there are other ways to do this but this is a very common one). After that websites save cookies to identify you and can match where you came from and what you bought.

You are indeed right that this is very hard to measure. It is somewhat easier on social media as advertisers often get detailed information about who got shown which ad at what point and how long it stayed on screen, if they moved their mouse over it, etc. Obviously the easiest indicator for an ads effectiveness is to measure how many people clicked the ad and then how many stayed on your webpage navigating around and then how many actually spent money and how much money they spent. But this does not give very accurate results as a lot of people who see an ad will ignore it initially but then talk about the product more frequently after having seen the ad which then might result in a sale through another channel later on. So you need to try to correlate the different ad campaigns with things like mentions, visits to your webpage, and eventually sales. And this can be very hard to do right.

There are a lot of metrics companies can use. Browser cookies will tell a company if you clicked on an ad, and whether or not you bought something as a result. Another tool is discount codes; give one code to one advertiser and a different code to another, and see which gets used more. You could do something similar with phone numbers or websites.

Depends on the ad and the businesses reason for the ad. Social media sites have reporting for businesses who advertise that can show different info. So one ad you might be trying to get as many people as possible to see it. While on another you may be tracking how many people click the ad or how many people click and purchase something.

TLDR; Platforms like Reddit and Facebook have tools that let you see performance.

Source: I work in advertising.

Run different variations and see which ones get higher clicks, higher conversions (lead sign up, purchase, etc)

There's simple reporting metrics, impressions, click through, dwell time, etc. Depends on the channel its on.

They also do market research. Talk to a sample of people, ask if they recall the ad, their impression of it, their opinion of the product or brand.

They may even run multiple ads to different audiences and compare the results to measure success.

Links on social media include unique identifiers that are passed through to analytic software. Fb click ids, google click ids, everyone has click ids.

When you make a purchase on a site they will tell facebook that your browser made a purchase so keep finding users like you.

The analytic software will use cookies, ip addresses, and other unique data about you. All of your transactions are tracked and you will be remembered when you return to the site.

But how do you know if it was effective, you can have a metric saying X amount of people saw your ad but doesn't necessarily equates to sales, how do you know the conversion between the amount of people that saw an ad and the sales that generated?