Kagura would need signature yes, and unless the beast is actively in use you would have to be almost right up their face with prolonged focus to pick the beast out anyway, as seen with Karin.
Sage mode, signature yes, however if beast isn't in use the ability to pick out the host anyway is far more questionable, because senjutsu hax isn't answer to everything.
Byakugan's a mess because there is no unified consensus of its 'insight', much less range. However I will daresay there is no way you'd find a host quickly, especially without signature and beast not being in use, since even if the eye has massive range its not like you are constantly aware of everything happening in that entire kilometers long range. Like a shorter range I might understand the 360 from, but there's no way your brain would handle constant awareness of every single detail at like 10+ KM range, especially when at several points in the series its been shown hyugas still focus on specific things at even longer ranges.
Be those as they may, difference is you need time/effort and most likely signature in all of the above. Telescope will find them no matter what, wherever they are, unless their chakras basically shut off (aka untrackable) or they're in a location somehow sensing proof otherwise.
.....okay, so when is a bijuu ever in active use in RP? Let say it is, how would the challenger post in such a way that it would work out. Would that person need to post RIGHT AFTER the Jink's post? Relatively after? Does the posting have to take place in the same 'area' of the RP (village board, fighting zone, etc)?
In regards to Byakugan, the last known range was 20 km based from Hinata in The Last: Naruto the Movie. You can read her wiki page for that. But everyone can agree it has at least 10 km range which covers an area greater than Brooklyn, NY. The point is that a village is not bigger than a city. >.>
Remember when Neji found those seals for the Five-Seal Barrier.
http://www.mangapanda.com/naruto/263/8https://youtu.be/LTJpPd0-hWo?t=367It took him like 10 to 15 seconds (based on animation time) to find 4 seals, shortest distance = 350 meters, longest distance = 800 meters.
Who's to say that a Hyuuga cant use his byakugan like a radar/sonar scanning section by section if "there's no way your brain would handle constant awareness of every single detail at like 10+ KM range"? You said yourself, 'hyugas still focus on specific things at even longer ranges' (Oddly enough, I only see this with Hinata while Neji and Hiashi did not show signs of needing to 'focus' - Naruto Chapter 613).
But honestly, you don't need a 'constant awareness of every single detail' because you specifically looking for something. And (I think) we all saw the movie Limitless. >.>
And realistically, getting a bijuu's chakra signature could be a tough task if you have not experience a bijuu before in anyway. And plus, correct me if I'm wrong, the bijuu's chakra signature would have to belong to THAT particular bijuu. Although funny enough, if clansmembers have similiar chakra signatures, then shouldn't the bijuu's chakra signature be similiar? They all came from the same place (ten tails). >.>
Except it can be, because just like I also pointed out in said thread, Naruto's circumstances in that moment were so very specific that there is no way to plausibly apply that as the universal truth to every host search via byakugan.
What circumstances?
And problem with telescope is that even if its supposed to give just a limited view and all like you say, people still won't actually use it like that. In some cases even they won't use it at all, rather they just say they have it so all the relevant info location-wise should be automatically accessible to them. Gets even fishier when claims are made it was used, but at PMs, dwellings, or somewhere else you can't actually see.
Well if you're already familiar with the village, then technically yes, you would instantly know the location. For example, if I use the telescope tech to find a bijuu and the bijuu in question is Kumo. As a shinobi that's born and raised in Kumo, I should know exactly where that location is, no?