If you want to get this done fast, the best way would be to put together a group of 8 people to boost respect giving. You can do it with 3 people (minimum required), but I'm not sure if that's time-efficient. My experience is with a group of 8.
Edit May 18, 2018: It seems that this can be done with only 2 people now (thanks ParadoxReal8 for the info).
This is what you should do:
1) One player is appointed as host and creates a Room (it has to be public). To do that, you go to Online > Rooms and press X to create a new room. You will be asked to enter a name for the room. My advice would be to choose a name that will not show up at the top of the list so that fewer randoms see it & therefore join in. In my case, I had a TA session set up to do this, so I used "ta session" as room name.
2) Once the room is created, the remaining 7 players go to Online > Rooms and look for that room on the list and then press A on the name of the room to join.
3) When all 8 players are in the room, the host selects the 7 players that joined (this is done one by one) and presses A on each of their names to challenge them. They will be added to a new list that is shown on the right (only the host sees this).
4) When all players have been added, the host selects any of the other 7 players on the list that was created on Step 3 above, and presses A to challenge that player. Make sure you choose "Survivor" (yes, it counts as long as it's a public room), as this is only one round per fight (you deal a little less damage in this mode, but it's still faster than two rounds dealing more damage). This will join everyone to the match (Finally!!)
5) After a while (it can take up to 1 full minute I think, or it seemed so) the lobby is created and you will see the fight screen on the left (larger) and a sort of snap screen on the right with all 8 players. It will be organized like this:
And now we're talking! Players 1 & 2 fight, and at the end of the fight, a wheel will appear for all 6 observers and they will be asked to give their respects. Make sure you select 10 each time (with the left stick and pressing A) and be aware that you only have a limited time to give respect. Selecting the number can be flaky, what worked best for me to prevent the stick from moving to other numbers was to hold it on the desired number and press A before releasing the stick. Respect points go to the winner of the fight only.
After this process, Player 1 will be moved to the bottom of the list and become Observer 6, Player 2 will become Player 1, Observer 1 will become Player 2 and everyone else will move one step up the ladder.
To make things easier, what we did was always have Player 2 win. That way you make sure that everyone will get some respect should connection or other issues strike.
I found the fastest character to be Kung Lao with his Forward + B kick, if done fast you will get the win at 77 on the counter. Each full round takes about 10 minutes if everyone uses him and no one is messing around. That would be 60 respect each in 10 minutes, or 360 each in 1 hour, so in 4 hours everyone can get 1000 respect & their achievement and even allow for some connection issues.
You or someone in your group are likely to experience connection issues at some point and be kicked from the lobby. If that happens, the player that dropped will be put in a lobby by himself and a waiting or connecting message will appear. In that case, you (the kicked player) should go back to the menu, go to Online > Rooms like before, find the room, join, press A on the name of any of the other players and press A again on the "Join Survivor" option. This should have you re-join the lobby and you'll be able to resume from there. If you were kicked when it was your turn to win and you hadn't been put on the Player 1 position yet, then you'll lose the respect from that fight (and only that fight). If this happens, I would recommend keeping the same method of Player 2 winning every time and make up for any connection losses at the end (see why below).
If after joining the room and pressing A on the name of any of the other players you get a message that says something along the lines of "waiting for connection" and the "join survivor" option doesn't appear, quitting the game and relaunching it might do the trick. One of the observers can also send an invite to the kicked player (I think this is done with LT, don't remember exactly). Or you might be out of luck sadly. You can always try and create a new room entirely, it would be up to the group and probably to be considered as an option depending on how far along you are.
Finally (yes!!), not everyone will hit 1000 at the same time, so when someone does get to 1000, you can enter AFK to make things faster. The AFK (Away From Konsole) mode will skip you when it's your turn to fight but you will still be able to give respect points, so this is really neat :) This is why you make up for any respect points lost at the end, and also because there might be other players that lose points further along the session.
I hope this is useful, let me know if you need clarification on anything here, as it's not difficult to boost but the getting ready process is rather cumbersome and a bit hard to detail.