I don't believe that any object other than dust or particles have been observed passing through our solar system. So the shooting stars you see are all from here. The giveaway would be the speed. The Leonids in November are among the fastet meterors. I believe they travel about ~ 70km / second relative to Earth. But that's still slow enough for a solar orbit. Anything coming in much faster would not be bound to the Sun, and would have to have come from outside the solar system.
I'd be willing to guess that extra-solar shooting stars do happen from time to time, but I do not believe that there's any documented observation of one.
It is possible that some of the comets that we see and some of the Trans-Neptunian Objects (TNOs) such as Sedna originally came from outside the solar system and were captured into Solar orbit when the Sun was young and possibly in a star cluster where it formed. Close passages of Brown Dwarfs and other stars would be much more likely then.
And since shooting stars are usually cometary debris, it's possible that many of the shooting stars we see originally came from outside the solar system, although they're currently trapped in a solar orbits.
Alessandro Morbidelli and Harold F. Levison investigated this possibility in their paper:
http://www.boulder.swri.edu/~hal/PDF/CR105.pdf
Using a formula Grant gave me a few weeks ago, I re-created their experiment of a 0.05 Solar-mass Brown Dwarf with a disk of particles ranging from 20 - 100 AUs passing 200 AUs from the Sun with a velocity of 1km/s at infinity.
I got the same results they did. ~40% of the Brown Dwraf's particles which orbited in the plane of the encounter were captured into Sedna-like Solar orbits.
http://orbitsimulator.com/gsimyabb/1.GIF
As the simulation begins, the Brown Dwarf system is closing in on the Solar System from a distance of just under 1 trillion kilometers.
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http://orbitsimulator.com/gsimyabb/2.GIF
The Brown Dwarf system closes its distance to the Sun. The green planet around the Sun is Neptune. It's there just to give some scale to the solar system. It won't help with particle capture.
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http://orbitsimulator.com/gsimyabb/3.GIF
As the Brown Dwarf system gets even closer, the Sun's gravity starts distorting the system.
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http://orbitsimulator.com/gsimyabb/4.GIF
The Sun sends a strong gravitational tidal force through the Brown Dwarf system. Objects outside the Brown Dwarf's Hill sphere are stripped away.
Some enter a Solar orbit.
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http://orbitsimulator.com/gsimyabb/5.GIF
After all the dust has settled, 8 objects which originally orbited the Brown Dwarf are now orbiting the Sun, some in Sedna-like orbits.