Bringing this thread back to life (maybe).
I agree with the statement that there is no right or wrong option here. Both flats and rounds sound good on a P-bass and the quest would rather be finding the right brand and tension that feels good under your fingers. As far as the sound goes I recommend recording yourself (live and studio) and listening back to what really is the practical difference between the two. A lot of times the 'zing' described by round-lovers is something that is mostly heard at home or really close to the speaker... and it often doesn't last that long. If that's what you're after, and I get the idea of liking that sound, be prepared for a life with buying a lot of strings and changing them pretty regularly.
On the other hand there's the alternative of going the 'flats-route', especially popular on a P-bass, that will give you a more relaxed approach...maybe spend a bit more when buying your set but then you often don't have to worry to much about changing strings for a long time. BUT... you won't get the zing and from time to time flats suffer from dead strings as well as rounds and having payed more makes it much more hurting to your economics and mental status.
So is there a middle ground here? I'd say you need to try for yourself because different basses seems to sound very different with different strings (did I mention different?).
Myself I have settled for about 3 different brands of flats where my go to flat is the Ernie Ball group II flats 50-105 because of their consistency and a tension that seems to fit my playing style very well. They are a safe bet if I want my basses to sound clear, warm and not get any issues with weird string buzz. They simply balance my basses up in a really good way. I should note though, that a couple of years ago I had some issues with a batch all of a sudden not being polished enough or simply not finished right. I tried to contact Ernie Ball about this but I was neglected and never got any answer from them. I hope that was a one time issue.
I also recommend La Bella flats for the warm traditional P-bass sound. They have a tendency of not being as even in production quality as the EBs and almost every set has one bad string and that's a shame since once you find 4 good La Bellas they will give you great sound and feel for a pretty long time... it just might take you a while.
GHS traditional flats are also good quality and not expensive but they are very much on the old school, no treble side of the spectrum and sometimes I feel like they just have been polished a bit too much and are very dead, but in a nice thumpy way, from the start. But if that's what you're after it's a good shortcut getting the GHS flats and you don't have to worry about string settling in as with most other flats. You're in vintage-thumpland right from the first tone
Ok. Now roundwounds. It's a different story and even more to chose from so I'll simply just shortly adress my favorite sets here. Rounds tend to die on you faster but what I've found is all three of the brands I'm mentioning here seems to my ears to still sound good even after loosing their initial top.. some of them actually better since the fundamental gets more solid when all the zing & twang is fading out.
Rotosound Swing bass SS 45-105
My favorites for years and I don't get what many refers to as them being so raw and noisy under your fingers. Ok, your fingertips will get black the first day of playing them but that nothing I personally have ever felt physically, I just see it visually. After a couple of days though they give me a great sound and they have that special mid presence that I have not found in any other roundwound (just listen to Jaco). Also they most of the time 'die' in a nice way and you could keep them on for quite a long time... they still give you good solid bass tone, maybe just not the zing.
Two more brands I like that are more elegant and mid scooped than Rotos but with a nice mid and defined body are DR Low riders SS 45-105 and Sadowsky Blue label stainless steels 45-105. Both very good quality rounds that will last longer than most and that has a good balance between zingy top, warm mid, and usable low end.
Last but not least the strings I mentioned has a character under my fingers that I like, both as far as tension goes and also as far as resistance (non-resistance). For instance the very popular D'Addario chromes to me is too sticky and harsh too long and I don't have the patience to wait for them to settle in and become what some describes as perfect. I'd then much rather go for the Ernie Ball flats which is a similar string but without the downsides of the Chromes. Nickel rounds in general to my taste sounds too much like you want them to sound acoustically but as soon as you use them in a band situation they just are too mellow and weak as far as the body of the note you're playing goes. I feel i just get lost in the total sound picture.
All this is of course one opinion among thousands, probably millions and for every theory there is a whole lot of contrary theorys.