Kedushat shevi'it for nectarine and peach pits

Answer

 One should try to eat as much of the fruit as possible (like children who lick the bits of chocolate from the wrapper or drops of ice cream from the container). That is, if during regular years you generally leave lots of fruit stuck to the pit, during shemitah you should try to eat as much as you can (you don't have to go crazy pulling out the little strands of fruit from the pit, but eat more to show you care).

After we do this, the small bit still stuck to the pit is insignificant and is secondary (tafel) to the pit itself.

If you left a significant amount of pit on the pit, this is considered an important amount and it has kedushah.

Pits

Some posekim maintain that since fruit pits are edible for some animals, they have kedushat shevi'it. However, the mainstream ruling is that pits that are not earmarked to feed animals, and when very few people would give such pits to their livestock, they do not have kedushat shevi'it.

They can be thrown directly into the garbage. If you have fruit or a pit with lots of fruit with kedushat shevi'it, it is forbidden to place it directly into your composter. First store it in the shemitah bin for several days, and when it is no longer edible you can then put it in the composter.

Conclusion

If you didn't finish the entire fruit and there is a significant amount of fruit attached to the pit – it goes to the shemitah bin. If you finished the entire fruit and there are a few strands of fruit stuck to the put – it does not have kedushat shevi'it and can be thrown into the garbage. If you didn't try to finish more than usual and there is fruit stuck to the pit, technically you can throw it away, but it is better to put it in the shemitah bin for several days.