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You can start trumpet vines from seed, but there are much easier ways to propagate this vine. Since you already have one trumpet vine, you'll have a much greater chance of success if you transplant a sucker, or take a vine branch that is touching the ground and simply tack it down to the soil (with a hair pin, etc.) until it roots.
There are several ways to plant the seeds, but they need around 60 days of cold treatment to ensure a successful germination.
In the Fall, you can sow untreated seeds directly outdoors. You can also sow seeds directly outdoors in the spring, but you'll need to stratify them first (nick the seed hulls to enable better water penetration).
You can also germinate seeds indoors in 15-30 days after a cold treatment. To do this, place the seeds on top of a growing medium in a container (don't cover them). Seal the container or place it in a plastic bag to retain humidity. Keep in the fridge and check for germination in 4-8 weeks.