Yes it is as legally binding as any document drawn up by a solicitor.....however you need to make sure that you have worded everything correctly and covered every single aspect.
If you have witnesses to your signatures then it is legally binding. You do know that most of these contracts can be "got out of" whether they are lawyer written or not though....so don't put too much faith in them.
A contract doesn't even have to be in writing to be binding - a verbal contract is just as binding. Anything in writing (and any witnesses to it) would be evidence of the terms of the contract and your intention to enter into legal relations.
As has been said...a verbal agreement is legally binding but can always be open to dispute as different parties only remember what they want to remember about the agreement.
I written agreement is far better, it doesn't need to be written out by a solicitor as long as it is signed by both parties & witnessed it is binding. The problem comes when there is a dispute. The contract has to be worded so that there are no ambiguities & the meaning of each section is clear & concise. Do not assume anything, everything must be in the contract. Do not use fancy language, use simple easily understood terms & don't use slang.
Definitely get a solicitor to draw up any contract.
A privately written or a verbal contract may look like it will stand up in court but to be honest a good solicitor will be able to dismiss one very quickly.
A written contract can be legally binding without a solicitor yes, but if the wording is not right, or there is an obvious loophole that the lay person would not detect then it could be easy for one of the parties to wriggle out of the contract. Get your solicitor to take a look.