I can give you the Italian answer - first of all, normally we don't use tomato paste to make sauce, but rather to add a tomato "kick" to recipes. Tomato paste is simply tomato puree that has been cooked down to a high degree of concentration.
A basic tomato sauce is made by
- making a soffritto with onion, carrot and celery (plus other flavors)
- adding tomato puree, or "pelati", more rarely fresh tomato
- cooking the sauce down until the taste and thickness is what you want
If you want to use paste instead of puree, the third step has to be omitted or greatly reduced in duration. Keep in mind that tomato paste has its own taste, and that taste will remain in the final sauce.
As the others have mentioned, it won't quite be the same ... however, it can be done, it just won't quite have the same flavor as there isn't a chance for the meat juices to transfer into the sauce, and some issues regarding mixing it back in with the cold pasta.
If I were going to try it, I'd brown the meat, remove some of the fat if it's a really fatty grind, then add some extra sauce (or even other liquid ... maybe dairy, for a classic bolognese) to let it simmer for a bit ... then toss the pasta in with it to reheat.
The extra sauce is the key, otherwise, the meat isn't going to blend in with sauce on the pasta, If you reheat the pasta in the fresh sauce, it should hopefully mix together more completely.
... but if I were to do it, and wanted to add a meat to an already cooked pasta, I'd probably not choose ground beef as a first choice; I'd probably cook up some sausages, cut it down to an appropriate size for the pasta you're dealing with, and mix that in, possibly adding some extra liquid when reheating the pasta.
Best Answer
My secret weapon is onion. Caramelize the onions first. This creates a natural sweetness. Always use ripe tomatoes; if you cannot then use canned. The canned are made from ripe tomatoes and tend to be a very good substitute.
Also, the celery and carrot suggestions are very much a good addition - you are making a classic tomato sauce when you include the celery and carrots, 1 part each celery and carrot to 2 parts onion. Add the onions first to get the caramelization.
I don't get too fancy with herbs, usually one good fresh herb does it for me, basil or sage are two favorites of mine.
There is a suggestion to add cream, why not use the parmesan end you may have in your fridge, it is dairy and allowed to slowly simmer in the sauce will lend a very nice flavor note.
Acidic sauces are usually reserved for nights when I make a puttanesca sauce - it goes with the capers and calamata olives I add to it.
Don't forget to season with salt and pepper.