I think you can assume it's open-pollinated unless the plant is isolated from others of its kind (or species it can hybridize with) and/or you hand pollinated them yourself.

Plants with common names aren't necessarily hybrids. Blanket flower, for example, is the common name for gaillardia (several species share the common name), but names like "Fanfare" is a hybrid/mutant variety name. Which is which you usually just learn from experience (not a painful one by any means) and by looking at the way the names are written down, usually. You will usually see the names listed like these, for example: Gaillardia "Fanfare", or Blanket flower "Fanfare".

As for how you can know if it's going to be a hybrid, or if it's going to come true, basically, you won't until you germinate them and they flower (or the leaves come out, in the case of hostas). Hostas generally won't come true, but there are some "rules of thumb" about some variegated varieties that I know of (but can't remember at the moment). When you trade, all you can really say is something like, "Open pollinated (usually assumed if from a garden), parent plant is like so-and-so". With hostas, you don't necessarily need to bother mentioning much. Most people know that hostas don't usually come true.

Otherwise, known hybrids don't usually come true, but that's not necessarily true either. Some do, and some do to a certain extent. Dierama "Blackbird" is a hybrid that is reputed to come true 90% of the time (I have seeds, but haven't tried it), for example. You can usually read here and there if it does come true. Else, assume it won't, if it's a hybrid.

Good luck, and have fun!

There are perhaps two sides to your question.

The first has to do with buying open-pollinated or heirloom seeds and the second has to do with saving seeds from open-pollinated plants.

And unfortunately the two aspects are not necessarily related.

Some species of plants naturally self-pollinate - that means their seeds are formed because the pollen of the same plant fertilizes the ovules (the egg).

Some species of plants naturally cross-pollinate - that means their seeds are formed because pollen from a different plant fertilizes the ovules.

Some species of plants form their seeds from both self-pollination and cross-pollination.

Often an open-pollinated/heirloom variety is grown in total isolation from all other varities of the same plant species. In this case it does not matter much whether the plant self-pollinates or cross-pollinates since all the nearby plants that can provide pollen are very similar. The seeds will all be very similar and produce plants of the heirloom variety. That is how commercial seed growers would produce an open-pollinated variety to insure purity of the seed.

But if you grow an heirloom plant near a different plant of the same species and they can cross-pollinate then the seeds (or some of them) will not be the original heirloom variety.

If you grow only one open-pollinated variety of each species then as long as your neighbours do not grow the same species your seeds are very likely to be pure.

If the variety you grow naturally self-pollinates (in a very high proportion) then your seeds are likely to be pure.

For most other seeds you could name the seed parent (but not the pollen parent) and indicate that the seeds were produced by natural open pollination as suggested above.

Both open-pollinated and hybrid varieties may have a 'name'. For example Brandywine tomato is an heirloom open-pollinated variety but Carnival tomato is a hybrid. In seed catalogs hybrids are nearly always identified as such and open-pollinated or heirloom are also nearly always identified.

Hosta, Irises, Lilies, Daylilies, Peonies and similar perennials are hybrids even when not so identified. That is because there are few if any open-pollinated varieties or strains of these species. However, you can grow the original species of these plants and they are likely to be open-pollinated and come true from seed. Occasionally, a mutant of the orignal species is good enough to grow in gardens and there are some of these that also may come true from seed.