In total, 99 were seized by police under the Dangerous Dogs Act, down 60 from 159 in 2015.
Figures were obtained from all Welsh police forces using a request under the Freedom of Information (FOI) Act.
National Police Chiefs' Council spokesman Deputy Chief Constable Gareth Pritchard said many people bought dogs without knowing their temperament.
North Wales Police said attacks rose from 158 to 246 in the area. In south Wales, it went up from 298 to 332 and in Dyfed-Powys' area the numbers increased from 45 to 53.
Gwent Police recorded number decreased from nine to three, in the figures obtained for BBC Radio Wales' Jason Mohammad Show.
Dep Ch Con Pritchard, who also works for North Wales Police, said: "We are concerned people are buying them on the internet.
"They are buying at car boot sales and buying at motorway service stations.
"They are not seeing the dog in its environment and understanding its temperament."
He added that if people are thinking of buying, they should "see the dog with the mother, see the temperament and understand the power and the attributes of the dog".
Despite the increase in attacks, the same FOI figures show the numbers of seizures under the Dangerous Dogs Act have fallen from 159 to 99.
In North Wales Police's area, the figure went down from 32 to 16, in south Wales, from 113 to 72, there were two seized in 2015 and 2016 in Gwent and in Dyfed-Powys, the figure dropped from 12 to nine.
Jayne Dendle, from Swansea, is a member of the campaign group Born Innocent, that lobbies against the Dangerous Dogs Act.
"Owners can just go out and buy any puppy, everything is totally unregulated and people don't necessarily see a dog as a lifetime pet and don't put the effort and energy they need into it," she said.
"Socialising it and exercising it. A dog is a big commitment and we need to put a great deal of time into that."
There were 634 reported dog attacks in Wales in 2016, up by 124 from the year before, according to latest figures.