If the other site is still in the formative (definition or commitment) phase, and you notice a potentially interested user, the best way to let them know on SE is in a chat. Sometimes, leaving a comment is appropriate, but this should be done very sparingly, as you'd essentially be spamming for the proposed site.
Other sites (proposed, beta or launched) often come up when discussing the topicality of a question. So watch for borderline questions whose closing is being discussed in comments, or clearly off-topic questions that would be appropriate for the site you're trying to pimp. If the site is beta or launched, suggest a migration that a moderator might carry out. You can also participate in topicality discussions on meta sites, or post on the “where can I ask questions that aren't X questions” meta thread (e.g. SO; English doesn't seem to have one, you could create it).