When do we know when we train hard enough?

I tend to know I am training hard when I am dripping sweat within 16 minutes of starting, and/or when my muscles ache the next day. Also when I am out of breath is a good indicator. I train alone other than once a week with a personal trainer.

Start sweating within 15 minutes of your workout,
(Tip: Rest short, have good technique)
When you feel sore next two three days.

it depends.

For very short, (an)aerobic workouts such as sprint repeats and one minute efforts, I know I'm at my limit when I feel I"m about to throw up. Happened a couple of times.

For long efforts (20+ min) on the bike (aerobic), I've had both the "pursuiter cough" that happens when you stop as the lungs fill with fluid that need to be cleared. Or having a bit of blood from excessive dryness in throat/nose.

Those are maximal efforts. I don't try them every time. For a normal day, try a few repetitions (either on the bike or resistance) until the quality of the work can't be maintained. So if your form is falling apart, you stop. If you're supposed to be going 25 mph and you can only do 22, you're cooked and done for the day.

Quality over quantity. TIP - when you're really at your limit or ignore what is your capability (e.g. starting a new activity, or coming back after a layoff), some coaching or direction can provide a better assessment.

EDIT - the talk test is a very good indicator. If you can communicate in short sentences, the work is hard but sustainable. That doesn't mean you have to train at that intensity (and definitely not all the time, you want easier and harder, breakthrough sessions). But it's an important benchmark.