Back many years ago, when we were on our first Panda (750cc Panda), although there was nothing really wrong except it was a bit rusty and I was worried the hex was starting to round a bit, I bought a, very cheap, new sump plug from an auto accessory shop. It did exactly as you describe, screwed in with no sign of tightening! It cost peanuts so I just chucked it and screwed the old one back in - which continued to give good service for more than 12 years!
The way these plugs tighten is that the plug is actually a tapered shape (I think it's the plug and not the hole? - but yes, I'm sure it's the plug) So the further you screw the plug into the hole the tighter it gets as the taper uses up the slack. I think the plug I bought - and probably yours - were/are straight sided, like a normal bolt, so have no hope of tightening. Probably buying one from the local main dealer or a trusted supplier (my local factor are good as are shop4parts to mention just 2) will ensure you get a "good" one. How far do you screw it in? 'till it's reasonably tight and I find this leaves the end of the plug just proud of the face of the threaded hole in the sump. I do find though that it differs on each vehicle. At present our Panda's is just proud but the Punto's identical setup has two or maybe three threads showing. I'm very keen on using a torque wrench on alloy sumps but this is steel and tapered so your chances of stripping it are just about zero.
We've had a Panda 750, Panda 1000, Panda Parade and our present Panda Dynamic Eco. My older boy has a Punto 1.4 8 valve. All have/had this type of sump plug and I've serviced them all. If the plugs are in reasonably serviceable condition they self seal due to the tapered nature of the plug. You don't need to tighten them excessively, just "Nice 'n tight". I don't use thread locker or sealant of any type and never had a plug loosen off or leak. If it worries you though there's no reason why you shouldn't use a little medium (blue) thread locker for peace of mind. Generally I would advise against plumbers thread tape (PTFE is it?) as little bits tend to get loose inside the sump and might get sucked into the strainer. I do find that if you tighten them very "enthusiastically", the next time you come to undo it it is almost as if it's locked itself in place. I have read on this forum a number of posts from people who have had trouble loosening them. I think what is happening is that, because the plug is tapered, if you substantially over tighten it, during tightening the oil on the tread makes tightening easy due to the lubricating film but the wedging action of the taper forces the oil out of the threads. Then it spend maybe a year in this "tight threaded" condition where most of the oil in the threads is forced out so now the actual thread forms themselves are as good as "dry". Now, at service time, you are undoing a thread which was well oiled when tightened but now dry and "grippy" as you try to slacken it.
All the best. Do let us all know if you've got - what I suspect to be - a straight sided plug. It would be interesting to hear from anyone else who has heard of this or actually found themselves with this problem.
Regards
Jock