In the UK, this is regulated by the Rehabilitation of Offenders Act. Under the act an applicant does not need to disclose any spent convictions for most roles.
Convictions become "spent" depending on the sentence given. Cautions and non-custodial convictions become spent most quickly. Custodial sentences of over 2½ years never become spent. The age at which a crime was committed also affects when a conviction becomes spent. Some roles, such as "teacher" are exempt and all convictions and cautions must be disclosed. — source
It is illegal, under the act, for a potential employer to ask about spent convictions, or to treat an applicant with spent convictions differently.
Getting comparable statistics is difficult as the statistics for different countries may use different samples (including or not those who receive lesser sentences or various forms of caution), measure different outcomes (re-arrest, re-conviction or re-imprisonment). And over different time-spans. Attempting to compare the UK with the USA directly suggests that re-offending rates are lower in the UK than in the USA.
Re-offending rates in the UK are at about 25% within 12 months (source). Using the wider measure of "re-arrest" suggests that about 50% of prisoners are re-arrested, though not all are subsequently convicted or imprisoned. This compares with about 60% in the USA.
There does not appear to be much controversy regarding the application of this law to adult offenders. There is debate on how it should apply to juvenile offenders source
In Europe, disclosure rules must be consistent with the European Convention on Human Rights, which gives a right to privacy, except as necessary for the protection of public good.