How do you A/B test by person (as opposed to device)?

Question

I understand the general concept of A/B testing (randomized controlled trial where you split users to see one version vs another and measure effect of something on a metric, e.g. click-through rate).

My impression is that the basic A/B tests are done at the user/device level, so if I'm viewing a website on my computer browser vs phone, I may see different versions. What's the best practice in accounting for that? Or do most folks in industry just live with the fact that users might actually be exposed to both treatment and control if they visit the website on two different devices?