First the good side. There are several species of deer you can hunt in WA State. Most likely if you start hunting deer, you will get interested in the 2 species of elk available too.
You will find blacktail deer on the westside of the Cascade mountains and Olympic Peninsula. Also known as the WETSIDE of the state. Forested lands are 43% privately owned. Predominantly by private forest companies, requiring a purchase of access pass from the forest companies. Motorized Access pass fees will cost
$300-$350 annually per person. Back in the 1970-1980's, access was free. Not anymore. But do the research and you will find some State DNR, a little BLM and some national forest access.
For mule deer, you need to research game management units in central, north on the east side of the Cascade mountains, SE and few areas of NE part of the state. Referred to as the DRY side. Focus on main national forest lands for public access. Again about 41% of eastern WA forested lands are privately owned. More BLM lands on the east side, State DNR lands as well.
Good populations of whitetail deer in the SE and NE parts of the state. However, lots of private lands in the prime whitetail units but good parcels of public national forest are available.
Sadly 50% of the positions on the WA game council are chaired by known non-hunter/activists. Most are appointed to the positions by the State Governor or his appointed Game Commissioner. The game commissioner himself admitted publicly that; "I am a hippie from San Francisco with no hunting experience or history thereof". Along with; "hunters are needed less to control game populations as we have healthy numbers of predators to do so:" And he does love wolves, with special 11-person committee to oversee the wolf populations. 6 of these persons are non-hunters, special interest group backing by PETA, Humane Society and other similar factions.
A sorry welcome for you to hunting Washington State.
Forest Landownership, Forest Inventory Analysis Annual Inventory
TOTAL / PUBLIC / PRIVATE
2014 Ownership Entire State (acres) 21,981,284 / 12,574,566 / 9,406,718
Western Washington (acres) 12,215,007 / 6,857,951 / 5,357,056
Eastern Washington (acres) 9,766,277 / 5,716,616 / 4,049,662