I actually only just re-read the relevant chapter last night, so for once I'm happy to do a quote  :)

 

"It is difficult, since we have bolstered the rumours that she [siuan] was executed", they have bolstered the rumours through propaganda from the Tower (scribes etc) but they don't need to walk the streets screaming she was executed. Furthermore, if enough sisters are told that she was executed by Alviarin etc then maybe there will be sisters bound by the oaths saying it; however I think its just the tower sisters trying to make the WT seem more stable to the outerworld, rather than sisters misleading sisters.

As far as runors go,

 

"Suian Sanche has been stilled and sentenced to death" would become "Suian Sanche has been excecuted" by the time it had passed three sets of lips.  That's the beauty of rumor in TV.  No one has to lie.