I would say a definite no.
There is a reason why the Missal says that Priest alone speaks, or Priest with Concelebrants. Believe it or not, it is in fact a Liturgical Abuse for you to do such things. Whilst it will not effect the validity or the licity of the Mass, it is merely disobedience to the Missal that has been developed, compiled and
approved by the Church. The Church is the Divine authority on the earth, in terms of doctrines and liturgy, it is the earthly voice of God’s will - as the Catholic Church was the one true Church established by Jesus Christ, the perfect man.
It is no sin, to do such things, but one must remember; the Priest alone can offer the Holy Mass, we can only participate. Yes, listen to the words and allow your soul to be opened to witness of the Sacrifice that lays before your very eyes, but there is no need for you to speak, when it is the Priests role to speak at this point.
Would it be weird if a Priest said “The Lord be with you” then joined in on the reply “And with your spirit” - Yes, it would be weird; because it is the Server’s (and the people’s) role to speak at that particular point.
We must stay in obedience to the liturgy; T
he Catholic Church is the Universal Church. Therefore we must enforce unity in our way of praying, maintain tradition and authentic faith.
Say the Black; Do the Red.
In magnam fidem,
Deus Salus Nostra
P.S. Sorry for the big Theological rant but what I say is important, and said in good faith and obedience.