The party says the move would reduce the temptation for sex offenders to abscond and make it easier for police to trace them if they do.
At present, the decision on whether to publish identities lies with the police.
Scottish Government figures obtained by the Tories last year showed that there had been 247 breaches of sexual offences prevention orders since 2008.
Conservative justice spokeswoman Margaret Mitchell said: "Given the potential consequences for public safety, sex offenders who breach their conditions should lose their anonymity.
"These are often dangerous and predatory individuals.
"Losing their right to anonymity would serve to protect the public by making them aware these individuals were on the move.
"It would also deter those sex offenders who may be considering breaching the terms laid down on the Sex Offenders Register."
A Scottish Government spokeswoman said: "This is an operational matter for the police and local law enforcement agencies which are responsible for monitoring sex offenders.
"The vast majority of sex offenders comply with their conditions but where an individual seeks to evade justice, their details can be published through a range of media.
"Each decision is an operational matter for the police, based on specific intelligence and a professional assessment of risk and the potential results of publishing or not publishing."