Peppermint sticks last up to 12 months before they lose their freshness.
The peppermint sticks will last longer than 12 months and still be safe to eat past 12 months but after 12 months they don't usually taste as good.
Also in the hot humid weather the peppermint sticks can lose the peppermint as it can melt off the peppermint sticks so it's best to refrigerate the peppermint sticks if your house is too hot.
But otherwise the peppermint sticks can be kept in the pantry or on the counter or just out at room temperature for a year and longer.
If you want the best taste from the peppermint sticks it's best to eat them before 12 months.