And are their problems with them?

Syncro 1 - generally not well regarded but neither is Syncro 2 and I think that works fine. I’ve never used Syncro 1.

I concur, it's first generation Syncro. The 2nd lever changed the mode between friction and index, which is performed by a pull and twist knob on Syncro II. Syncro II also added a 2nd set of detent selectors for more positive engagement of the notched insert.

Syncro has a worse reputation than Syncro II but that is because Campagnolo was initially trying to make it index with their older design friction derailleurs and various freewheels and chains. By the time of Syncro II, they were designing index derailleurs and having their partners manufacture compatible chains and freewheels, so Syncro II appeared to work better, though there's little actual difference in the index performance of the two lever sets.

Synchro = meh.

Synchro II = meh II.

If you have any experience with indexed shifting, you won't like this. It's as bad as it gets.

I have two bikes with Syncros. One with a first gen Syncro, with a six speed freewheel, and the other with Syncro II with a seven speed freewheel (Regina FW and Sedis narrow chain on both systems.). Oddly, the system with the older first gen levers, actually works quite well, but the one with the later Syncro II's, not so well,....yet. Might have something to do with the chain and freewheel combination I chose for the seven speed system.
I suspect the Campagnolo engineers were in such a hurry to get an indexed system out to catch up with Shimano that they did not bother to test the system on the road for enough miles. And instead of designing it correctly in the first place, they tried to do sort of "Band-Aid" type of corrections to the system, which didn't really fix them (Hello Delta brakes?).
If you insist on keeping the Campy Gruppo complete that might mean you need to keep the Syncro and you can't get them to work that well?.,......just always keep it in friction mode....

What they've said. Synchro I worked to a point, and I learned quickly that if it came to me working, to leave it alone. The chain and freewheel used were important factors, for sure, as well as matching the right RD and indexing ring.

I had a Pinarello Montello built in Humpy Wheeler's bike shop (yep, the speedway guy is a bike fan) in Charlotte, Synchro II, and it was fine. It probably had under 50 miles on it, and I sold it to a collector before the decals fell off. I doubt it's been ridden since.

I also bought a Sychro I group and put it on a Cinelli, with limited success, finally opting for Ergo 8sp instead. The various mechanics the owner took it to advised on freewheels, RD's, A/B settings, chains, etc. It ended up on a Canopus and remained finicky.

A while ago, I had a conversation with one of the former mechanics that worked where the Pinarello was built. He said the best thing to do with a Synchro group was a) if it works, leave it alone, and b) if it doesn't, replace it.

Great looking stuff, and if dialed in, no problem, but getting it right was always beyond my abilities.

I think Syncro reputation suffers in that people will try to throw a drivetrain together with a somewhat random collection of parts and then get frustrated when it doesn’t work. Campy tried to make a system that would work on many drive trains, with mixed success. I would suspect one would get the same level of frustration trying to mix/match bits in a same era Shimano group. FWIW my Merckx Pro came with first gen C Record with Dura Ace shifters. Shifting was meh but I put some Syncro 2 on it and it’s much better now.

Does the friction work well enough to just leave it in friction mode?

It does, if the friction discs in the levers are not too worn to hold a setting. just like any other friction system.The Syncro II will be easier to leave on friction mode because the knurled knob on the lever does not get in the way of shifting once it is set to friction. Also, as the function setting knob sits into well defined notch(es) within the lever body when set as friction that there is really no way for it to come off that setting other than deliberately pulling up the knob (that has quite a firm spring on it) and turning it to the indexed setting or vice-versa. Not the so with the first gen Syncro, as the second lever on the RD shifter can easily be bumped during shifting, turning it back to indexed setting which it defaults to when the two levers are aligned.
The bike that I have the Syncro II on is currently set at friction mode so at least I can ride it. It will go back to indexed mode once I figure out how to make it shift better.

I've had good results using Synchro in both 6s ad 7s configurations.
Usually the old bikes that I had bought needed first to have the shifter indexing ring, springs and pivot lubricated at least with a few drops of oil to move snappily so as to hold indexing position precisely. The rear-most cable "loop" of housing also needs to be lined and clean inside, preferably with modern "compressionless" housing fitted and no unneeded additional length(!).

Another big variable (as already mentioned) is the choice of freewheel and chain. Modern chain, especially using Shimano 9s chain with 7s freewheels makes a huge difference, enough even to compensate for indexing error caused by the result of using say a (S.I.S.-7-spec) SunRace 7s freewheel with appropriate (to most of these Synchro derailer's capacities) 13-25t range. So actually one can get great Synchro shifting using such a somewhat wrongly-spaced freewheel even in indexing mode, and there may be room to experiment further with gearing range (note that S.I.S.-7 cog spacing uses a wider spacing between the 2nd and 3rd-smallest cogs).
There were "compact" (narrow spaced) 6s freewheels and perhaps indexing rings to go with them, but I would avoid that setup since there are no Uniglide or Hyperglide freewheels in that configuration, and the otherwise-excellent Suntour U6 freewheels have the asymmetrical spacing issue that I don't think any Synchro ring was designed to deal with.

I mentioned the Shimano 9s chain because:
A) they are narrower than Synchro-era chain so minimize unwanted contact with the next-larger cog that one is shifting to, and
B) they have a modern tooth-grabbing profile with the largest difference between outer width and pin width of any 9s chain on the market,
both of which allow for a more-forgiving shifting action without any unwanted running noise.
Additionally, modern chain is more flexible, so somewhat simulates the effect of a floating top pulley.

^^^^^

I agree, I use modern 8 speed chains on mine and think it makes a big difference, throw a ramped freewheel in there and even bigger improvement.

I like vintage bikes for the simplicity. I may just swap these out for regular friction shifters.