Your monitor profile tells the computer how to translate from standard color values into levels your monitor needs to show the correct color. Nobody else needs an image tailored to your monitor. If you had two monitors, even you wouldn't want an image encoded for just one specific monitor. Software should take care of it, converting from the image's monitor 1 profile to a neutral standard, then converting again into a profile to display on monitor 2, but like all conversions, it can cause loss of precision.

You will want to export images a standard profile. Typically that would be sRGB unless the image need a wider gamut, and then pick another standard space like adobeRGB or DCI-P3.

Most web browsers these days understand color profiles, so you shouldn't have much trouble with others viewing your images if you use something other than sRGB. You may have trouble getting correct prints if you stray from sRGB unless your print-shop explicitly mentions using other color profiles.