We have had poor experience with Appcelerator's push service. I don't think that Appcelerator is focused enough on push notifications specifically. On numerous occasions, their platform would mysteriously underdeliver (maybe getting to only 30% of our audience), and getting diagnostics on these events was like pulling teeth (like a week of back-and-forth with their support team to get them to investigate the problem). They never really provided much guidance in terms of best practices for client-side integration of their SDK.
We recently moved to Pushwoosh, and have been much happier with their platform. They are extremely knowledgeable on all things push-related, and they have given us an impressive amount of support in our questions about arcane topics like push tokens, ios do-not-disturb mode, android background services, using multiple push providers, etc.