I cringe when I see someone seeding tomatoes and then dumping the seeds. All part of my obsessive compulsion and need to grow them.. I have to fight the urge to scoop them up and run. Not every tomato makes it to the list.. just the ones after a quick taste that I find worthy.
Some of the grocery store tomatoes are specifically bred to handle being transported, uniform size, uniform ripening... in other words.. taste plays little role. Sad really.. as it is one of those fruit so common in our diets, but we are so trained to select the perfect looking specimen and miss the fact the weird and sometimes gross looking ones taste divine.
The resurgence of heirlooms and obscure strains are making a comeback. As more people declare themselves foodies..and chefs push the envelope.. they often lead the way in selecting food according to taste. For the tomato... this is a wonderful thing.
Tomato seeds are covered in a gel coat that naturally contains germination inhibitors.
Removing the gel coating is generally done by means of fermentation. Super simple.. it is basically adding the seeds (and some of the tomato juice, or a bit of water) in a container with a lid.. closing it up and allowing it to sit in a spot out of direct light. I use a glass jar and give it a shake maybe once a day. You are waiting for the fungus and bacteria to work their magic. With a glass jar.. I can keep an eye on the seeds and see when the gel coating has come off. It takes a few days, but when I see the coats have slipped off.. I shake the jar really well and and a bunch of water. The viable seed will sink and the tomato bits & fungus will float. I pour that off, add more water and repeat the process. Then I pour them out onto a coffee filter and move them around every few hours so they don't stick. You don't want to put them on paper towel to dry as they will cement themselves to it. Whatever surface you decide to dry them on.. make sure you move them as they are initially drying and try to keep them spaced out a bit with decent air flow.
It is a smelly process. If you know for sure that the tomatoes are from healthy plants.. you can skip some of the scent in this endeavor by putting just the tiniest bit of dish soap in the mix. I mean we are talking a scant drop to a full quart.
If the plants are diseased.. it would take more drastic measures to try to save them. There are treatments like hot water, bleach dip, tri-sodium phosphate (TSP) to name a few. If it is a virus and it is in the embryo, not much really you can do. Tobacco Mosaic Virus is one of those pesky pathogens.
This is also why you want to make sure you clean all equipment between each use in a quick bleach dip or hydrogen peroxide. Better to be cautious than potentially infect further batches because you didn't take 2 minutes to clean the jar or strainer properly.
Paste tomatoes take more effort as they are by design selected to be meatier and have less seeds.If you like drying tomatoes to preserve them or even sauces.. when you save seeds keep in mind the traits you want and select accordingly.
In a dry location with good air flow and minimal light it takes roughly two weeks to dry the seeds. The way you test them is to bend a seed in half (easier said than done with some of those tiny ones!) and when it is dry enough.. it will snap in half. If it bends, it has too much moisture still.
Make sure you write in permanent ink on the coffee filter what type of seeds they are unless if your whole approach is to be surprised until they fruit.
If you can.. try to save seeds from several plants. Tomatoes are one of those lovely seeds that can store for a very long time. Keeping them in a paper envelope in an airtight jar which is stored out of light.. at 6 years you still should have 80% or better germination. Even at 12 years you should have over 50% germination. Like with peppers, tomato seeds brought down to 3% moisture can be stored in the freezer for decades.
The key is keeping moisture low, controlling temperature and darkness. From just a few tomatoes, you can in fact get enough seeds to last you many years. (Usually they are self fertile.. wind moving the plant is often the main means of pollination... however... insects can cross pollinate them. This year I have particularly diligent bumblebees and I watch them almost daily hit up the tomato flowers.)
It is easy to do and economical. Currently a rough average is about $3 to $3.50 for 25 seeds.