Dogs were among the first settlers in America.

There were two dogs on the Mayflower (1620) and probably a few on the Abigail (1628) along with livestock for the Naumkeag settlement headed by Governor John Endecott. While dogs were in North America before colonization, they were mostly in the West. With the advent of colonization, native dogs were gradually almost entirely replaced by European ones.

Early on in Salem, wolves were plentiful and problematic. Especially during the winter months, wolves would target the domestic animals of the settlers, resulting in major losses of livestock necessary to the colony. So bad was the situation, that in 1635, the General Court set a bounty of five shillings for each wolf and one shilling for each fox killed that would be paid from the town treasury.

There were few dogs in the colony that could protect livestock from these threats, so the bounty was necessary. The fact that there were a few is evident from Felt's Annals when he wrote of the Quarter Court imposing a fine and imprisonment on John Sweet for having killed a dog that belonged to Captain Endicott in Endicott's yard in 1637. Captain Endicott owned the dog to protect his livestock. Apparently, Sweet mistook the Governor's dog for a wolf. The fine he received was later rescinded. The breeds most often mentioned early on were mastiffs, greyhounds and English spaniels.

For a number of years bounties were placed on wolves and foxes. These bounties were rescinded when attacks lessened only to be re-imposed again with renewed wolf attacks. The usual method for killing wolves was with traps and weapons. The bounty varied depending on the severity of attacks and whether the wolf was killed in town or outside. In 1640, the usual bounty of 10 shillings was raised to 40 if hounds were used. This was probably a testament to the value of dogs and the dangers in using them to hunt wolves.

In 1645, Salem voted to come up with a better solution which was to import six or four braces (pairs) of hounds from England. These dogs would be paid for by the town. This apparently worked well, since the General Court in 1648 gave selectman in every town the authority to procure and import as many hounds as they saw fit and assign the dogs to whoever was best suited to care for them. They further ordered that no one could own any dog without permission of the selectmen.

When King Philips War broke out in 1675, it was recommended that dogs be used to hunt Native Americans in areas where fighting was occurring. In 1706, the legislature passed an act encouraging the raising and breeding of dogs for better security along the frontier. The following year, the government paid troopers and huntsmen for training dogs for use along the frontier of Middlesex.
In 1756, the legislature, in response to a widespread bout of distemper that killed large numbers of dogs and cats, required that selectmen appoint buriers for dead animals to ensure that all animals were buried at least two feet deep to avoid contagion. Those owners who neglected burial were to be fined 40 shillings and the cost of burial for each infraction.

In 1773, during an epidemic of small pox, Thomas Heather cried an order calling for killing cats and dogs least they spread small pox. It's not clear if this order was heeded.

In 1784, a State Ways and Means Committee proposed a tax on dogs.
They estimated the dog population in Massachusetts at 30,000. The tax was never passed. At this time, dogs were widespread in the states. A painting of George Washington bidding farewell to General Lafayette shows the president and his family with his two dogs, a spaniel and a border collie. As Colonial expansion continued, more working dogs were imported on a regular basis.

In the interest of equal time, as well as the fact that family members own cats, I'd be remiss if I didn't mention that domestic cats also immigrated from Europe during the 1600 and 1700s where they were aides in curbing the rodent population. They were frequent passengers on ships to help curtail rodents and early on had opportunities to immigrate, which they did in large numbers.