dnd 5e – As a DM of a group of 4 players, would it be appropriate for me to run a private 1-to-1 session so that a PC can act in secret?

In fact, we have done it several times in a campaign in which I am playing. Players or DM occasionally carry out relevant side missions for a few selected players, whom we then find time to play on another day of the week. This prevents it from disrupting the group as a whole by preventing them from playing, while allowing the secondary search to be carried out.

The smaller group also helps make the interaction much faster. One session was literally the DM and I on a 15-minute car ride, playing in the theater of the mind. Another was just my assistance in finding another player's side, and it only took us half an hour. Another of mine took about an hour, and was very similar to what you described earlier (me, the rogue, in a secret search for thieves).

So yes, this is completely possible and normal. As I mentioned earlier, simply do your best to avoid interrupting the other players, either by making them feel through them, or by having to skip what would normally have been a group session.

However, keep in mind that by allowing this for a character, you can open the door for others to request the same. This does not mean you have to do it, especially if you only intended this to be a unique opportunity to help the other player connect better with his character. But be prepared to know how you intend to respond and handle those requests.

Impact to the main story.

This part depends a little more on you as DM and on how you want this to affect your world. In the same way that background stories can (and should) affect the overall plot to increase character participation, they can and should also present stories. Ideally, you want to avoid the consequences that separate the agency from other players in your part of the story, but that does not mean that it cannot be relevant.

For example, in my search for the side of thieves I mentioned earlier, the completion of the task led me to an underground criminal network throughout the city. This allowed me to help other members of my group as an informant, who could obtain information about missions and objectives that we, as the whole group sought, and helped increase the dynamics of our group. This had an effect on the story by helping to improve my investment in it and provided another resource for the other players, but it took nothing away from others.

Side note: XP handling

As NautArch suggested in a comment below, one thing to keep in mind is how this could affect the gain of XP. Depending on the leveling method you use (standard XP, milestone, 3 pillars, etc.), you should be careful that these side missions do not give someone an unfair advantage over their peers. In our campaign, side missions generally only grant XP to those who were behind the others, often with the search itself as an intentional way for that player to catch up.

As someone who used to attend each session and was the leader in XP, I often agreed with my DM to resign completely in order to simply enjoy the adventure, or in other cases I would join another person's secondary search (if they were fine) with him) as a way to help them level their character faster Again, everything will depend on their group and how the leveling works, but as long as the side missions don't give them a special or unfair advantage, they should be good