Is it maybe a spittle bug? I have lots of problems with them on my rosemay and basil plants. I just wash them off and use the plant anyway. They look bad, but don't do a lot of damage.

Since you use the word 'sticky' to describe your problem, I would guess that you have an infestation of mealybugs, not uncommon to rosemary. A population of spittlebug larvae are almost always described as 'spit'.

A picture would help with ID, however.

In the meantime, you can help manage the pest by applications of insecticidal soap, which can be used on herb plants. I get infestations of spider mites on my rosemary, which makes them sticky as well.

FYI, moniquest now thinks that 'spit' is the right word to describe the culprit! There's no doubt, then, that he/she has an infestation of spittlebugs. We don't usually worry too much about these little guys, as ltcollins suggests. Strong sprays of water would remove that spittle nest, making them easier prey for predaceous insects....or an application of insecticidal soap if the population is very high.

Thanks for the input...just spotted this a few days ago!