In additional to the potentailly privacy issues mentioned above, localbitcoins (or whatever other similar service you may use) can potentially censor your transaction, which may be difficult for you to detect. They can do this for any reason, or for no reason, and because of the privacy issues mentioned above, they have the potential to censor your transaction for a reason specific to *you*.
How (and why) should an online service (like localbitcoins) 'censor' a transaction just because it didnt get broadcastet by OP's node, but someone else's node?
I don't see a single reason why someone would 'block' transactions just because 'a 3rd party' broadcastet it.
Im not even sure if thats completely transparent (who exactly broadcastet this TX first; IP maybe? )?
A "
pushtx" service could censor a transaction by simply not broadcasting said transaction, possibly while saying the transaction was actually pushed, or by saying the transaction is somehow invalid. When a user uses a pushtx service, the only way the network sees the subject transaction is vis the node(s) of the pushtx service.
Why this type of service would want to censor a particular transaction could be a number of reasons. Maybe the service does not like the person, or maybe it is suspicious of the history of certain inputs that are being spent, or any other arbitrary reason.