Not necessarily. There are many Eastern Orthodox, Anglicans, Lutherans, Old Catholics, etc, who are quite happy where they are. It’s sad to know there is very little open communion between us all.
When a Catholic receives the Eucharist, they are proclaiming, among other things, the following according to the Catechism:
1369 The whole Church is united with the offering and intercession of Christ. Since he has the ministry of Peter in the Church, the Pope is associated with every celebration of the Eucharist, wherein he is named as the sign and servant of the unity of the universal Church. The bishop of the place is always responsible for the Eucharist, even when a priest presides; the bishop’s name is mentioned to signify his presidency over the particular Church, in the midst of his presbyterium and with the assistance of deacons.
1370 To the offering of Christ are united not only the members still here on earth, but also those already in the glory of heaven. In communion with and commemorating the Blessed Virgin Mary and all the saints, the Church offers the Eucharistic sacrifice. In the Eucharist the Church is as it were at the foot of the cross with Mary, united with the offering and intercession of Christ.
1374 The mode of Christ’s presence under the Eucharistic species is unique. It raises the Eucharist above all the sacraments as "the perfection of the spiritual life and the end to which all the sacraments tend."201 In the most blessed sacrament of the Eucharist "the body and blood, together with the soul and divinity, of our Lord Jesus Christ and, therefore, the whole Christ is truly, really, and substantially contained."202 "This presence is called ‘real’ - by which is not intended to exclude the other types of presence as if they could not be ‘real’ too, but because it is presence in the fullest sense: that is to say, it is a substantial presence by which Christ, God and man, makes himself wholly and entirely present."203
1376 The Council of Trent summarizes the Catholic faith by declaring: "Because Christ our Redeemer said that it was truly his body that he was offering under the species of bread, it has always been the conviction of the Church of God, and this holy Council now declares again, that by the consecration of the bread and wine there takes place a change of the whole substance of the bread into the substance of the body of Christ our Lord and of the whole substance of the wine into the substance of his blood. This change the holy Catholic Church has fittingly and properly called transubstantiation."206
1378 Worship of the Eucharist. In the liturgy of the Mass we express our faith in the real presence of Christ under the species of bread and wine by, among other ways, genuflecting or bowing deeply as a sign of adoration of the Lord. "The Catholic Church has always offered and still offers to the sacrament of the Eucharist the cult of adoration, not only during Mass, but also outside of it, reserving the consecrated hosts with the utmost care, exposing them to the solemn veneration of the faithful, and carrying them in procession."208
How could a non-Catholic receive the Eucharist in the Catholic Church if they do not hold these same beliefs?