How tight should cycling shoes be?

IMO you want your shoes to be a "generously" large fit when at rest. Not so large they slide around on your feet but still biggish while remaining comfortable.

Why? Your feet will subtly swell during exercise, making a good fit into a tight fit. That's why your slightly large fit becomes good after some time.

Naturally this assumes your socks are the same thickness as you'd wear when riding. Avoid test fitting shoes with big woolly socks, unless that's what you ride in too. Upshot here is that your winter and summer feet may vary in width by a noticeable amount.

Length of foot doesn't change as much, because the only flesh is around the heel, in the toes and between the joints. Instead its more about width of foot, across the top more where the skin is more flexible/elastic.

This is one of the reasons why velcro closers and boa dials and other tension-adjustments have grown in favour - noone's going to retie their shoelaces while riding, but you can reach one hand down and tweak a strap/knob without stopping.

You control the angle of your skis through your ski boots. Any slack between your foot and the boots compromises your directional control. Therefore the boots need to be really close fitting. My toes are right at the end of my ski boots, and it hurts when I put them on. However, you control the direction of your bike through the handlebars, and it is possible to ride a bike fairly fast wearing flip flops. Therefore your cycling shoes just need to be a comfortable fit.

This answer is geared especially to those buying the shoes remotely, as one would do in 2020.

The pair of shoes has arrived, and you want to determine whether it's the right size.

First of all, your feet are almost surely not identical. Pay more attention to the larger of the two. There are two useful tidbits of information to know about your feet: 1- which foot has bigger bones, and 2- which foot has bigger bulk (muscle mass, or "meat"). The answer to both questions may not lie in the same foot.

Now, if your toes touch the front of the shoes, tighten the straps. Are you able to tighten the straps so that you have adequate blood supply and get your toes off the front? If yes, you've found your right size.

Perhaps the most important part of the answer is this: You do want to tighten the straps as much as you can (while maintaining blood flow). If you don't, you will lose a small bit of efficiency. Your foot will wobble up and down during your stroke. Ideally, you do want to finish your down-stroke and immediately after find yourself able to start the up-stroke (and vice-versa). A "lag" is not just irritating if you want to go fast. It's also disturbing because, for just an instant, you will not have power, when you were hoping that clipless pedals & shoes provide power throughout the 360 degrees.

The straps themselves are likely quite adaptable. If your feet do swell during your workout, as Criggie suggests, you'll have to readjust in the middle of your ride. This is presumably unsafe during a ride, which is likely why one of the upgrade features for the more advanced models on the market is the ability to adjust through a dial rather than through reclosing the velcro straps.

I'm a performance-oriented cyclist. I prefer to have my shoes fit firmly around my foot. I would consider that if my foot is squirming around in my shoe, I might get blisters, and I would be losing some power. However, your feet won't move as much if you don't frequently do hard efforts, and you might have a preference for a slightly looser fit. Also, with velcro straps or adjustment dials (e.g. the Boa system), you can adjust the amount of tightness/looseness a bit to account for preferences and potentially for feet swelling during a ride.

I have relatively limited running experience, but subjectively, I'd say my cycling shoes fit slightly tighter than my running shoes (albeit my competitive running experience is a lot more limited than my competitive cycling experience, and I might have steered towards tighter fitting runners for races). I'm afraid I can't compare to ski boots. Compared to the hiking boots I wear daily in winter, I'd again say that my cycling shoes are noticeably tighter. For reference, my two different brands of cycling shoe (Giro and Shimano), my Oboz hiking boots, and my Brooks running shoes are all EU size 40, although another person's experience might differ in a different size range.

To my knowledge, cycling-specific socks will be synthetic and they tend to be relatively thin. Cooler weather socks for standard cycling shoes (i.e. not winter boots) are available, like DeFeet's Wooleator, but these are noticeably thinner than, for example, wool hiking socks. I do wear the latter in my winter cycling shoes, and I believe it is standard practice to size up if you get a pair of winter-specific shoes. Hiking socks would not fit in my summer shoes. To the best of my knowledge, synthetic materials and many types of wool are better at dissipating sweat than cotton, which is desirable no matter the operating temperature.

As to shoe sizing, my experience with cycling shoes has been that manufacturers will often state the length of the shoe last (that's the foot form used to make the shoe) in centimeters or millimeters. The Japanese sizing usually corresponds to this length (in cm), if the manufacturer states it anywhere. Some manufacturers round to the nearest 5mm, or they depict the last lengths on a bar chart, i.e. they don't give you the actual length. You can measure your foot length. Either based on that measurement or on your experience with a previous cycling shoe model, you can get some sense of what last length you need, and look that up on the corresponding manufacturer chart.